Constructive quantum field theory — Supplement to Lectures 6–7 (notes)

This supplement contains the detailed proofs which Lectures 6–7 cite or condense. It is reference material: the lectures can be read without it. The results are recent, joint work with S.-J. Meyer + AI. Unless stated otherwise, notation is that of the lectures.

Where each result is proved.

• Free resolvents, Lemma 10.1: Lemma 19.1.

• The coefficient construction, Proposition 11.2: Section 20, in particular Proposition 20.6, with the rooted calculus and the trace comparison. The dependence on the boundary value is Proposition 20.8.

• The explicit counterterm, Theorem 11.3: Section 22.

• Global coercivity and the source bounds, Lemma 12.1 and Proposition 12.2, with the term-by-term ledger: Section 21.

• The facts of (12.5), (12.8)–(12.9) and the bound on \(\mathfrak{D}\) used in Lemma 18.1: Lemma 21.2.

• Consistency of the coefficients, Section 18.1 of Lecture 7, and Lemma 18.3: Sections 23–31.

• Results removed from the lectures, which are not needed there: regularity at finite resolution, the proof of the limit FBSDE by compactness and the Riccati equation, the density at a fixed scale, and two earlier proofs of uniqueness: Section 32.

The auxiliary scale \(q_{\ast}\) is as in the lectures. In Sections 19–31 it is the lowest scale of the action; below it the action is frozen (Section 12.4 of Lecture 6).

19Elementary kernels

We use the notation of Lecture 6: \(K_q = q - \Delta_N\), \(C_q = K_q^{- 1}\), \(G_q (y) = (K_q + \lambda y^2)^{- 1}\), \(H\) for the inverses of \(K_q + D^2 \Gamma\), \(M_q = \mu_q I + \Theta_q\) for the quadratic path, with \(\Theta_q = \Theta^{[3]}_q\) for the final one and \(M^{[j]}\) for the level masses of Section 20.4, \(\kappa_q\) for the clip, \(\varrho_q\) for the coordinate correction, and \(\text{Ctr}_E T\) for contraction. Recall \(g_q = \lambda q^{- 1 / 2}\) and \(r_q = (q / \lambda)^{1 / 2}\), so that \(\lambda r_q^2 = q\). Generic convolution kernels are denoted \(E, E'\), and the letters \(L, L'\) denote arbitrary kernels. Distance \(d\) is torus distance.

Forces and their primitives.

There are two scales, which must not be confused. The reference mass \(q_0 > 0\) has arbitrary size; it fixes the covariance \(C = C_{q_0}\) of the measure, and it is the lower end of the range of Lemma 19.1. The auxiliary scale \(q_{\ast} \geqslant \text{max} (1, q_0)\) is large, depending on \(\lambda\) and on the bound \(B_0\) below; all the operators of Sections 19–21 after Lemma 19.1 are used at scales \(q \geqslant q_{\ast}\) and do not involve \(q_0\). So from Lemma 19.3 on we assume \(q \geqslant 1\). Constants may depend on the coupling and on finitely many derivative orders, but not on \(N\) or on the scale interval. For a kernel, \(\|C\|_{\text{row}} = \sup_x \int |C (x, z) | \mathrm{d} z\) is the row norm (Section 10.1 of Lecture 6); it bounds \(\|C\|_{p \to p}\) for every \(p\) when \(C\) is symmetric.

Proof. The heat integral \(C_q = \int_0^{\infty} \mathrm{e}^{- qt} \mathrm{e}^{t \Delta_N} \hspace{0.17em} \mathrm{d} t\) proves positivity, the row mass and the operator norms. Killing the lattice random walk at rate \(\lambda y^2\) proves the first part of (19.2). The second part is \((K_q + \lambda y^2) 1 = q + \lambda y^2\).

The symbol of \(- \Delta_N\) is \(4 N^2 \sum_i \text{sin}^2 (\pi n_i / N)\), with \(k = 2 \pi n\), comparable to \(|k|^2\) on the centred frequency cube. Thus \(a_2 (q) = \sum_k (q + c_k)^{- 2}\) and \(\text{Tr} C_q = \sum_k (q + c_k)^{- 1}\), with \(c_k\) this symbol, \(c_0 = 0\). Counting the \(N^3\) modes gives the bounds \(N^3 q^{- 2}\) and \(N^3 q^{- 1}\), with universal constants. Nonzero modes have \(|k| \geqslant 2 \pi\). Splitting at \(|k| = \sqrt{q}\) bounds \(\sum_{k \neq 0} (q + c|k|^2)^{- 2}\) by \(Cq^{- 1 / 2}\) for all \(q > 0\), and \(\sum_{k \neq 0} (q + c|k|^2)^{- 1} \leqslant CN\). The zero mode contributes \(q^{- 2}\) to \(a_2\) and \(q^{- 1}\) to the trace: for \(q \geqslant 1\) these are at most \(q^{- 1 / 2}\) and \(1\), and for \(q_0 \leqslant q < 1\) they are at most \(q_0^{- 3 / 2} q^{- 1 / 2}\) and \(q_0^{- 1}\).

Let \(q \geqslant 1\). A dyadic shell is a smooth Fourier cutoff supported where the frequency magnitude is comparable to \(L\), with \(L\) ranging over powers of two; together with a low-frequency cutoff these form a partition of unity. Such a shell of radius \(L \leqslant CN\) has kernel bounded by \(C_m L^3 (q + L^2)^{- 1} (1 + Ld)^{- m}\). This follows by summation by parts; on the final shell use the periodic sine symbol. Summing gives \(C_q (0, z) \leqslant C \text{min} (d^{- 1}, q^{- 1} d^{- 3})\) off the diagonal. For its weighted square split the radial integral at \(q^{- 1 / 2}\): \(\int_0^{q^{- 1 / 2}} r \hspace{0.17em} \mathrm{d} r\) and \(q^{- 2} \int_{q^{- 1 / 2}}^{O (1)} r^{- 3} \hspace{0.17em} \mathrm{d} r\) are both \(O (q^{- 1})\). Discrete annuli give the same bound above the lattice kinetic scale; the diagonal has zero distance weight.

Let \(q_0 \leqslant q < 1\). Separate the zero mode, whose kernel is the constant \(q^{- 1}\): the kernel of \(C_q - q^{- 1}\) has multiplier \(1_{k \neq 0} (q + c_k)^{- 1}\), and that of \(C_q - q^{- 1} - (C_1 - 1)\) has multiplier bounded by \(c_k^{- 2} 1_{k \neq 0}\), which is summable uniformly in \(N\). Hence \(C_q (0, z) \leqslant q^{- 1} + C + C_1 (0, z)\). Inserting this and the case \(q = 1\) gives \(\int d (0, z) C_q (0, z)^2 \hspace{0.17em} \mathrm{d} z \leqslant C (q^{- 2} + 1) \leqslant C (1 + q_0^{- 1}) q^{- 1}\).\(\Box\)

Remark 19.2. (The zero mode) The dependence on \(q_0\) in Lemma 19.1 is genuine and comes only from the constant mode, whose eigenvalue of \(C_q\) is \(q^{- 1}\). On the one-site lattice \(a_2 (q) = q^{- 2}\) and \(\text{Tr} C_q = q^{- 1}\), so that the bounds \(a_2 \lesssim q^{- 1 / 2}\) and \(\text{Tr} C_q \lesssim N\) cannot be uniform as \(q \to 0\). The constants are universal for \(q \geqslant 1\). Since \(q_{\ast} \geqslant 1\), no constant below depends on \(q_0\) through this lemma.

For a symmetric centred convolution kernel \(\Theta\), that is, with \(\Theta 1 = 0\), put

The diagonal is determined by centring. We never need its absolute row norm. Define the convolution algebra and module norms

Here a translation-invariant kernel is identified with its displacement function \(E (z) = E (0, z)\), all integrals are in that displacement, and \((E \ast E') (z) = \int E (z - u) E' (u) \hspace{0.17em} \mathrm{d} u\) is convolution. Products \(EE'\) inside a single spatial integral instead mean pointwise products. The triangle inequality for distance, Minkowski, and Cauchy–Schwarz give

Also \(\|C_q^j \|_{\mathcal{H}_q} \leqslant C_j q^{- (j - 1) - 1 / 4}\); for \(j > 1\) use the heat integral bound \(C_q^j \leqslant C_j q^{- (j - 1)} C_{q / 2}\). We also use \(\|C_q \|_{\mathcal{A}_q} \leqslant C / q\). Indeed the unwrapped lattice walk has second moment \(6 t\), so its expected torus distance at time \(t\) is at most \(\sqrt{6 t}\). Integrating against \(\mathrm{e}^{- qt} \hspace{0.17em} \mathrm{d} t\) bounds the first moment of \(C_q\) by \(Cq^{- 3 / 2}\).

Proof. A shell of radius \(L\) and multiplier size \(a_L\) contributes at most \(Ca_L \text{min} ((Ld)^2, 1) \left( 1 + \sqrt{q} / L + \sqrt{q} d \right)\) to the first norm. Use two derivatives of its kernel when \(Ld < 1\), and translates otherwise. Put \(\Lambda = \sqrt{q}\) and \(t = \Lambda d\). Let \(\vartheta_{\Lambda}\) be a smooth even Fourier multiplier equal to one below frequency \(\Lambda\) and zero above \(2 \Lambda\), with the usual scaled derivative bounds. Decompose

The last two terms have shell sizes \(q^{- 2} L^2\) below \(\Lambda\) and \(L^{- 2}\) above it. After dividing by \(q^{- 1}\) the sums are

Splitting at \(x = t^{- 1}\) bounds each by \(Ct\); for \(t \leqslant 1\) the only borderline high-shell term is \(t^2 (1 + | \log t|) \leqslant Ct\). The first term in the decomposition is bounded by \(Cq^{- 1} t\) too. Above the lattice frequency take \(\vartheta_{\Lambda} = I\); nonzero lattice distances have \(t \geqslant 1\), and the delta kernel has the same bound.

Now use the signed identity \(C_q \Theta = \frac{1}{2} \int \Theta (z) \delta_z^2 C_q \hspace{0.17em} \mathrm{d} z\). The receiving identity \(G \Theta = (I - G \lambda y^2) C_q \Theta\) and \(\|I - G \lambda y^2 \|_{\infty \to \infty} \leqslant 2\) give the row bound. The form bound follows from the commuting Fourier multipliers of \(C_q\) and \(M\) and the row bound on \(C_q M\).\(\Box\)

Remark 19.4. (A necessary endpoint modification) The estimate \(\| \delta_z^2 C_q \|_1 \lesssim d (0, z)^2\) generally has a logarithmic loss at kinetic order two. We use one distance power and pay \(q^{- 1 / 2}\) instead. The recursion below produces exactly the compensating first-moment bound. This choice also rules out an unrecorded Laplacian in weak kernel limits: on \(d < \delta\) a smooth second difference is bounded by \(C \delta d\).

20Renormalisation with two correction steps

This deterministic section assumes no interacting field estimates. It contains the proof of Proposition 11.2 of Lecture 6.

20.1Tensors and the iteration

The kernels of field derivatives will be called vertices. A physical \(k\)-tensor \(T\) represents a multilinear form by

A slot means one argument \(x_i\). Fixing a slot as a root, and integrating all others absolutely, defines the rooted norm

For a translation-invariant tensor define its first spatial moment by

All derivative tensors are symmetric in their slots. A local derivative is allowed to contain physical deltas: \(\delta_x (z) = N^3 1_{x = z}\) has integral one. This convention prevents hidden lattice-volume factors when slots coincide. On the field ball \(\mathcal{B}_q\), the set of \(y \in H_N\) with \(\|y\|_{\infty} < 2 r_q\), put

Thus \(\textsf{S}_k\) controls the derivative everywhere in the field ball, whereas the spatial moment is required only at the zero field. The scalar bound \(\textsf{S}_k (f)\) is distinct from the posterior covariance \(S_q\) of (10.2). For a symmetric \(k\)-tensor \(T\), contraction of two slots against a kernel \(E\) means the physical integral

For tensor measures the same formula is interpreted after testing the remaining slots. Slots joined to inverse kernels in such a product are called operator slots; the remaining ones are external slots. The seed is the mass-subtracted logarithmic determinant, (11.6) of Lecture 6,

The square root and logarithm are defined by diagonalising the positive finite-dimensional operators; in particular this is a finite matrix logarithm. The subtraction removes its quadratic field term. It is nonpositive with zero constant and quadratic jets, and

For an even action define \((\Pi f) (y) = f (y) - f (0) - \frac{1}{2} D^2 f (0) [y, y]\); this removes the constant and quadratic Taylor coefficients, also called its low jets. For a quadratic path \(M_q = \mu_q I + \Theta_q\) (a scalar and an even, translation-invariant, centred kernel), whenever the displayed inverses exist, define the primitive of the trace map (11.7) of Lecture 6 (its field gradient is that map with \(\ell = Dh\))

Its quadratic jet at zero is, by evenness, the operator \(D^2 \mathcal{T}_q [M, h] (0) =\mathfrak{S}_q [M, h]\) of (20.8) below. For a translation-invariant operator \(L\) write \(L = \langle L \rangle I + L^{\circ}\), with \(\langle L \rangle\) its constant-mode eigenvalue and \(L^{\circ}\) centred. Given a scalar path \(\mu\), the recursion has three nonlinear functions and two centred kernels, as in (11.8): with the level masses \(M^{[j]} = \mu I + \Theta^{[j]}\) and \(\Theta^{[1]} = 0\),

Integrals are at fixed field, and the field balls increase with scale. The integrals are improper at infinite scale; their convergence is proved in Proposition 20.6. No derivative-losing infinite iteration is used.

20.2Two elementary receiving estimates

All kernels in this subsection are physical kernels, with normalised spatial measure. A closed cycle means the product of alternating vertex and inverse kernels, integrated over all joined operator slots. An external slot is left unintegrated until the rooted norm is taken. A marked factor is simply a selected factor that carries a difference estimate or a spatial-distance weight. Define

Schur's inequality gives \(\|L\|_{p \to p} \leqslant m (L)\) for all \(1 \leqslant p \leqslant \infty\). Also \(\mathfrak{l} (LL'), \mathfrak{l} (L' L) \leqslant m (L') \mathfrak{l} (L)\): on one side use the \(L^2\) operator bound for \(L'\), and on the other use Minkowski and its absolute row or column mass. These statements do not require positive kernels.

Proof. Choose the external root in \(T_1\). Integrate all external slots of the other vertices in absolute value. Their two-slot marginals \(\bar{T}_i\) have both Schur masses at most \(s (T_i)\), by symmetry. The open chain between the two operator slots of \(T_1\) starts and ends with an inverse. Its kernel is bounded by

Integrating the rooted \(T_1\) gives (20.5). Absolute kernels were used throughout; no cancellation or positivity of \(H\) was assumed.

For the spatial assertion, translation invariance identifies every rooted integral with its full spatial average. If the marked pair is in one vertex, charge its distance to that vertex's \(v_1\). Otherwise follow either arc of the closed cycle between the two marked external slots. The triangle inequality charges each piece to a within-vertex displacement or an inverse edge. A vertex charge gives a two-slot marginal with weighted row and column masses at most \(s (T_i) + \sqrt{q} v_1 (T_i)\). For edge charges the two arcs contain an inverse each. After the intervening vertices have been collapsed, write their convolution chains as \(E_1, E_2\). The closing integrals obey

Use one inverse in each \(\mathcal{H}_q\) norm and every other inverse and collapsed vertex in \(\mathcal{A}_q\), using (19.3). There are \(p - 1\) remaining inverses. This gives (20.6); there are at most \(Cp\) distance charges. The identical proof treats marked factors and the version with one fewer inverse. The weighted estimate uses \(q^{- 1 / 2}\); the sharper unweighted factor \(\mathfrak{l} (H)^2 \leqslant Ca_2 (q)\) is retained separately when needed.\(\Box\)

Remark 20.2. (The vertex must pay for its own spatial extent) Take the nonlocal fourth vertex \(T (x_1, x_2, x_3, x_4) = 1\). Since \(C_q^2 1 = q^{- 2} 1\), its contraction is the constant two-slot kernel \(\text{Ctr}_{C_q^2} T = q^{- 2}\). Consequently

On a lattice replace the integral by its normalised sum. Thus closing two free inverses cannot by itself give a \(q^{- 1 / 2}\) spatial gain for an arbitrary nonlocal vertex. This is why (20.6) retains every within-vertex moment.

Proof. For the diagonal map the positive majorant kernel is \(|L (x, a) L' (a, x) |\). Its row and column masses are bounded by \(\mathfrak{l} (L) \mathfrak{l} (L')\), using Cauchy–Schwarz in \(a\) and \(x\), respectively.

For the tensor contraction its majorant is

For its row mass first integrate \(a\), obtaining at most \(\mathfrak{l} (L) \mathfrak{l} (L') s (T)\). For its column mass put \(\bar{T} (u, v) = \int |T (x, u, v) | \hspace{0.17em} \mathrm{d} x\). Symmetry makes \(\| \bar{T} \|_{2 \to 2} \leqslant s (T)\), and therefore

Schur's inequality proves the assertion, including both endpoints \(p = 1, \infty\). For an input in a contracted slot, bound \(E\) by \(c\) and collapse the unused slot of \(T\); symmetry again supplies both Schur masses. An input in the open slot multiplies a function bounded by \(cs (T)\). These proofs also apply to diagonal vertex measures: the physical delta integrates to one, so it introduces no extra lattice factor.\(\Box\)

20.3The finite derivative budgets

Proof. Let \(H = (K_q + M_q + D^2 (U + h) (y))^{- 1}\), and \(\hat{H}\) the same with \(\hat{h}\). Taylor's formula, evenness, and the zero quadratic jet give \(\textsf{S}_2 (h) \leqslant Cg_q q\) and \(\textsf{S}_3 (h) \leqslant Cg_q q / r_q\); the same holds for \(\hat{h}\), and for their difference with \(g_q\) replaced by \(\varepsilon_q\). Thus the hypotheses control every Hessian vertex used in differentiating the inverse, including those with zero or one field derivative. Together with the local quartic they give \(s (D^{j + 2} (U + h)) \leqslant C_j qr_q^{- j}\) for \(0 \leqslant j \leqslant k\).

More explicitly, \(D^k H [u_1, \ldots, u_k]\) is the sum over ordered partitions \((P_1, \ldots, P_p)\) of the index set \(1, \ldots, k\) of

Here two slots of each vertex are operator slots, and \(u_P\) fills its remaining slots. Taking the trace closes this chain. Lemma 20.1 gives its rooted absolute bound

The factor \(a_2 (q)\) comes from the two \(L^2\) inverse rows, and each of the other \(p - 1\) inverses costs \(C / q\). Symmetry gives both Schur bounds for every collapsed vertex, also at nonconstant fields. The row Neumann series preserves the row and row-\(L^2\) bounds: use (19.5) for \(M\), and the just proved \(\textsf{S}_2\) bound for the nonlinear Hessian. All constants depend on the finite list of individual derivative bounds above.

Telescope the two partition sums. A changed vertex contributes \(\varepsilon_q\) by the difference budgets. A changed inverse equals \(- H (D^2 h - D^2 \hat{h}) \hat{H}\) and contributes the same factor, since its extra inverse costs \(q^{- 1}\) and its Hessian costs \(\varepsilon_q q\). More precisely, group that product as one marked inverse; its \(m\) and \(\ell\) bounds are \(C \varepsilon_q / q\) and \(C \varepsilon_q \sqrt{a_2 (q)}\). This proves the unweighted part of (20.7). The worst vertex uses \(k + 2\) derivatives.

For a quadratic inverse difference put \(\rho = | \delta \mu | / q + q^{- 1 / 2} v_1 (\delta \Theta)\). At a nonconstant field, (19.5) and the row Neumann series give \(\|H \delta M\|_{\infty \to \infty} \leqslant C \rho\). Thus \(\delta H = - H \delta M \hat{H}\) has absolute row mass \(C \rho / q\) and row \(L^2\) bound \(C \rho \sqrt{a_2 (q)}\). Transposing and exchanging the two backgrounds supplies the column bounds. Apply the same cycle estimate with this marked inverse. One must not replace this argument by an absolute row bound on \(\delta \Theta\) itself.

At zero field all inverses are the convolution \(R_0 = (K + M)^{- 1}\). The centred cancellation and the convolution Neumann series give

The identity \(\delta R_0 = - R_0 \delta M \hat{R}_0\), with \(\|C_q \delta M\|_{\mathcal{A}_q} \leqslant C \rho\), gives the same two bounds multiplied by \(\rho\) for \(\delta R_0\). Equation (20.6) now proves the first-moment part of the trace comparison, including changes of \(M\). Within-vertex distances use exactly the individual moment budgets in the statement.

For the seed, a derivative of order \(k \geqslant 4\) of the log determinant has \(p \geqslant 2\) inverse factors and \(p\) vertices: the local potential \(\lambda y^2\) has only first and second derivatives. The same cycle cut has \(p - 2\) intermediate inverses and gives \(C_k q^2 a_2 (q) r_q^{- k} \leqslant C_k g_q qr_q^{2 - k}\). At zero its degree \(2 j\) term is \((- 1)^{j + 1} \text{Tr} (C_q \lambda y^2)^j / (2 j)\); the same two-chain argument proves its spatial bound. Taylor gives \(\textsf{S}_2 \leqslant Cg_q q\), \(\textsf{S}_3 \leqslant Cg_q q / r_q\). Finally (20.2) says that \(\mathcal{T}_q [0, 0] - \partial_q \mathcal{L}_q\) is purely quadratic. Consequently

The first difference has a marked seed vertex and factor \(g_q\); the second has a marked quadratic inverse and factor \(| \mu | / q + \eta_q (\Theta)\). This proves the stated initial comparison.\(\Box\)

Remark 20.5. (Small Hessians alone do not suffice) The higher individual bounds in this lemma are necessary (compare Exercise 11.3 of Lecture 6). On the one-site lattice take \(q = \lambda = 1\), \(M = 0\), and fixed small \(\alpha, \delta > 0\). Put

Both corrections have zero quadratic jet and uniformly small Hessians on \(|y| < 2\), while all derivatives of their difference through order six are bounded independently of \(n\). Nevertheless, expanding \((1 + y^2 + h_n'' (y))^{- 1}\) to order four gives

Thus a trace Lipschitz estimate cannot follow from Hessian smallness and difference bounds alone. The example rescales to any \(q, \lambda\): put \(y = r_q s\) and replace each correction \(h (s)\) by \(qr_q^2 h (y / r_q)\). The displayed trace derivative is then multiplied by \(q^{- 1} r_q^{- 4}\) and still diverges with \(n\) at fixed \(q\). In our recursion the needed higher individual bounds follow at each finite step from the seed and the previous increment; they are not extra assumptions on the field.

20.4The coefficients, with a boundary value

Retain the quadratic jet and set, with \(R_{0, q} = (K_q + M_q)^{- 1}\), as in (11.9),

Let \(\mathcal{B}= [- B_0 \lambda q_{\ast}^{1 / 2}, B_0 \lambda q_{\ast}^{1 / 2}]\), an interval of boundary values, not to be confused with the field ball \(\mathcal{B}_q\). For \(b_{\ast} \in \mathcal{B}\) the scalar mass solves

Here \(\mathfrak{S}_q [M, h]\) is a deterministic operator-valued source, the linear coefficient of the loop force at zero field, and \(\beta_q\) is the constant-mode eigenvalue of the final one. The identity operator is written \(I\) when its space is clear. Translation invariance makes the constant mode a scalar. The centred parts of the sources at the first two levels define \(\Theta^{[2]}\), \(\Theta^{[3]}\) in (20.4), integrated from infinity; only the scalar of the final level is imposed, from \(q_{\ast}\). The linear force is \(M_q := M^{[3]}_q = \mu_q I + \Theta^{[3]}_q\), so that \(\partial_q M = \beta I +\mathfrak{S}_q [M^{[2]}, h^{[2]}]^{\circ}\), \(\mu_{q_{\ast}} = b_{\ast}\) and \(v_1 (\Theta_q) \to 0\) as \(q \to \infty\). The bare mass is \(\mu_N = \text{lim}_{q \to \infty} \mu_q\), as in (11.11), whose existence is a fixed-lattice assertion proved below. Note that \(q_0\) does not occur in (20.4), (20.8) or (20.9): these operators are built from \(K_q\) and the level masses alone.

Give scalar paths on \([q_{\ast}, \infty)\) the norm

Proof. Here every \(\Theta\) is even and centred. At each fixed \(N\) the norm \(\| \cdot \|_{\ast}\) defines a complete weighted supremum space of continuous scalar paths; restrict to the closed subset \(\mu_{q_{\ast}} = b_{\ast}\). It meets the ball of radius \(R\), since \(|b_{\ast} | / \left( \lambda \sqrt{q} \right) \leqslant B_0 (q_{\ast} / q)^{1 / 2} \leqslant B_0\) for \(q \geqslant q_{\ast}\). Work provisionally with radius \(R \geqslant 1\) and \(Rg_{q_{\ast}}\) smaller than a fixed constant.

The recursion on a given scalar path. Apply the trace lemma by a finite induction, with the level masses. The seed supplies its individual bounds through order ten, and \(\Theta^{[1]} = 0\). The first comparison, \(\Pi (\mathcal{T}[M^{[1]}, \mathcal{L}] - \partial \mathcal{L})\), with \(M^{[1]} = \mu I\), is the initial comparison of Lemma 20.4 with \(\varepsilon_q = C (g_q + | \mu_q | / q) \leqslant CRg_q\); it constructs \(h^{[2]} - h^{[1]}\) through order eight, and adding the seed gives the individual bounds for \(h^{[2]}\) at these orders. The centred source of the first level, \(\mathfrak{S}_q [M^{[1]}, \mathcal{L}]^{\circ}\), contracts the fourth seed vertex with \(R_{0, q}^2\), whose supremum is \(Cq^{- 1 / 2}\); the distance stays in the vertex, so its first moment is at most \(C \lambda^2 q^{- 3 / 2}\) and \(v_1 (\Theta^{[2]}_q) \leqslant 2 C \lambda^2 q^{- 1 / 2}\), that is \(\eta_q (\Theta^{[2]}) \leqslant Cg_q^2\). The local quartic contributes only to the scalar. The second comparison, \(\mathcal{T} [M^{[2]}, h^{[2]}] -\mathcal{T} [M^{[1]}, h^{[1]}]\), changes the vertex by \(h^{[2]} - h^{[1]}\) and the mass by the centred kernel \(\Theta^{[2]}\): write it as a vertex comparison at the mass \(M^{[2]}\) plus a mass comparison at the primitive \(h^{[1]}\), both covered by Lemma 20.4, the second with \(\eta_q (\Theta^{[2]})\) replacing \(\varepsilon_q\). Both have relative size \(g_q^2\), so it constructs \(h^{[3]} - h^{[2]}\) through order six and hence the individual bounds for \(h^{[3]}\). The centred source changes by the same two insertions: contracting the fourth vertex of \(h^{[2]} - h^{[1]}\) with \(R_0^2\), and the fourth vertex of \(h^{[1]}\) with the marked difference of \(R_0^2\), whose supremum is \(C \eta_q (\Theta^{[2]}) q^{- 1 / 2}\) by the marked inverse bounds of Lemma 20.4, gives a first moment \(Cq^{- 1 / 2} g_q^3\), and integrating from infinity \(\eta_q (\Theta^{[3]} - \Theta^{[2]}) \leqslant Cg_q^3\). These in turn permit the final fourth-order defect comparison. At each step the Hessian Schur norm is \(Cg_q q\), and every level mass lies in the coefficient ball, \(| \mu_q | / q + \eta_q (\Theta^{[j]}) \leqslant CR (g_q + g_q^2)\), so the inverses exist before the next trace is used. The centred kernels enter only through their first moments, absorbed in the resolvents by (19.5); they are never cycle vertices. In this induction the trace lemma and

prove the two increment bounds. Here are the details at the improper endpoint. Put \(F_s^{[j]} = \Pi (\mathcal{T}_s [M^{[j]}, h^{[j]}] - \partial_s \mathcal{L}_s)\). At the first step the initial comparison gives, for \(4 \leqslant k \leqslant 8\),

For \(F^{[2]}\) through order six, telescope it as \(F^{[1]} + \Pi (\mathcal{T}[M^{[2]}, h^{[2]}] -\mathcal{T}[M^{[1]}, h^{[1]}])\). The additional term has the smaller factor \(g_s^2\) instead of \(g_s\). All these powers are integrable. In the moment part of the integral, \(\sqrt{q} \leqslant \sqrt{s}\) permits the same domination for \(\mathcal{N}_{k, q}\). The projected integrands have zero constant and quadratic jets and are even. Taylor's formula therefore bounds their derivatives of order zero through three, on each fixed bounded field set, by the integrable fourth-order bound \(C_R \lambda^3 s^{- 2}\) times the corresponding field powers. Thus differentiation under the improper integral is justified through order eight at depth two and six at depth three. No unknown terminal jets or interacting moments are being imposed at infinity.

For fixed \(q\) and any compact subset of \(\mathcal{B}_q\), the same bounds hold in a common neighbourhood in field and scale, since \(\mathcal{B}_s\) increases with \(s\). The lower-endpoint derivative is therefore legitimate, and \(\Pi \partial_q \mathcal{L}_q = \partial_q \mathcal{L}_q\) gives

Moreover \(\textsf{S}_4 (h^{[j]}) \leqslant C \lambda^2 q^{- 1 / 2}\) and the zero low jets show that each \(h_q^{[j]}\) and its first three field derivatives vanish as \(q \to \infty\) on any fixed bounded field set.

The scalar fixed point. Contracting fourth tensors with the kernel \(R_0^2\), whose supremum is \(Cq^{- 1 / 2}\), gives for the source \(\mathfrak{S}_q =\mathfrak{S}_q [M^{[j]}, h^{[j]}]\) at every level \(j\)

In the centred part the distance stays in the fourth tensor. These bounds give \(\| \partial_q M\|_{\text{row}} \leqslant Cg_q\) and, for the scalar map \(\mathcal{A}\) defined by the right side of (20.9),

Here is the order of the constants. With \(R \geqslant 1\) and \(Rg_{q_{\ast}}\) smaller than a fixed constant, the inverse Neumann sums are bounded, for example, by two. The first increment has size \(C (1 + R) g_q^2\), the next has one further \(g_q\); after adding the seed, the individual derivative budgets have constants independent of \(R\) in this regime. Thus the source bounds above have a fixed constant, and the output bound reads \(B_0 + C_{\ast}\), with \(C_{\ast} \geqslant 1\) independent of \(R\) and of \(B_0\). Enlarging \(C_{\ast}\) if necessary, the centred outputs obey \(v_1 (\Theta^{[j]}_q) \leqslant C_{\ast} \lambda^2 q^{- 1 / 2}\) with the same constant, so every level mass lies in the ball \(\|M^{[j]} \|_{\ast} \leqslant R\) of (23.1). Now put \(R = B_0 + C_{\ast}\) and then choose \(q_{\ast}\) to satisfy the provisional smallness requirement.

For two scalar paths \(\mu, \hat{\mu}\) in the ball with the same boundary value \(b_{\ast}\), put \(d = \| \mu - \hat{\mu} \|_{\ast}\). The difference is a scalar insertion of relative size \(| \delta \mu_q | / q \leqslant g_q d\) at every level. Through the finite recursion, the trace comparison with mass changes gives \(\mathcal{N}_k (\delta h^{[j]}) \leqslant Cg_q^2 qr_q^{2 - k} d\) and, integrating the centred sources from infinity, \(\eta_q (\delta \Theta^{[j]}) \leqslant Cg_q^3 d\). Contracting with \(R_0^2\) as above, the scalar source changes by \(| \delta \beta_q | \leqslant Cg_q^2 d\). The scalar integral in (20.9) starts from the same value, so

After normalisation this is at most \(Cg_{q_{\ast}} d \log x / \sqrt{x}\) with \(x = q / q_{\ast} \geqslant 1\); the supremum of \(\log x / \sqrt{x}\) is \(2 / \mathrm{e}\). This is a contraction estimate on the whole half-line, with no prescribed bound on the bare mass. Increasing \(q_{\ast}\) once more makes it less than \(d / 2\). The closed ball is complete, so Banach's theorem gives the solution. Every choice here depends on the coupling, on \(B_0\) and on the finite derivative budget, not on \(N\) or on a probability moment order.

The defect. Evenness gives \(D^2 \left( \frac{1}{2} \text{Tr} H \right) (0) = - \frac{1}{2} \text{Ctr}_{R_0^2} D^4 \Gamma^{\text{o}} (0)\). The scalar mass equation supplies exactly the constant mode of this quadratic jet, while \(\partial_q h^{[3]} = \Pi \mathcal{T}_q [M^{[2]}, h^{[2]}]\) and \(\partial_q \Theta^{[3]} =\mathfrak{S}_q [M^{[2]}, h^{[2]}]^{\circ}\). This proves the defect identity. Apply the trace bound to the last increment to obtain the fourth-derivative estimate, with \(\varepsilon_q \leqslant Cg_q^3\) from (20.11) and \(\eta_q (\Theta^{[3]} - \Theta^{[2]}) \leqslant Cg_q^3\). The centred kernel \(\mathcal{E}^c_q\) has no tadpole contribution, since the local term is a multiple of the identity; it is the centred part of \(\text{Ctr}_{\delta (R_0^2)} D^4 h^{[3]} (0) + \text{Ctr}_{R_0^2} D^4 (h^{[3]} - h^{[2]}) (0)\), up to the factor \(- \frac{1}{2}\). The row norm of a contraction is at most the supremum of the kernel times the rooted norm of the tensor, and that of a centred part at most twice that of the operator. With \(\sup | \delta (R_0^2) | \leqslant Cg_q^3 q^{- 1 / 2}\) and \(\sup |R_0^2 | \leqslant Cq^{- 1 / 2}\), this gives \(\|\mathcal{E}^c_q \|_{\text{row}} \leqslant Cq^{- 1 / 2} (g_q^3 \lambda g_q + \lambda g_q^3) \leqslant Cg_q^4\); the first moment follows in the same way, the distance staying in the vertex.\(\Box\)

From now on \(q \geqslant q_{\ast}\) whenever \(\bar{\Gamma}_q\) occurs, and all constants may depend on \(\lambda, B_0\) but not on \(N\) or on \(b_{\ast} \in \mathcal{B}\).

Proof. Keep the sharp finite-lattice bubble in the unweighted trace bounds: \(\|R_{0, q}^2 \|_{\infty} \leqslant Ca_2 (q)\) and \(\textsf{S}_4 (h_q^{[3]}) \leqslant C \lambda^2 q^{- 1 / 2} = C \lambda g_q\). Contracting the fourth tensor in (20.8) therefore gives the first inequality, including both Schur masses. Since \(\int_{q_{\ast}}^{\infty} a_2 (s) \hspace{0.17em} \mathrm{d} s = \text{Tr} C_{q_{\ast}}\), its scalar integral is absolutely convergent, and (20.9) gives the bound on \(\mu_N - b_{\ast}\). The Fourier symbol bound gives \(\text{Tr} C_{q_{\ast}} \leqslant C \sum_{|k| \leqslant CN} (1 + |k|^2)^{- 1} \leqslant CN\). For the upper tail use \(a_2 (s) \leqslant N^3 s^{- 2}\). Finally every nonzero lattice displacement has distance at least \(N^{- 1}\). The absolute off-diagonal row mass of \(\Theta\) is at most \(Nv_1 (\Theta)\); centring bounds the diagonal by that same mass. This proves the last estimate. For the one-site lattice \(\Theta = 0\).\(\Box\)

The leading \(N\)-dependent part of \(\mu_N\) is \(- \lambda \text{Tr} C_{q_{\ast}}\), and it differs from \(- \lambda \text{Tr} C\) by a quantity bounded in \(N\). The next quadratic contraction contains the logarithmic two-loop mass divergence. Keeping its full running kernel avoids an incorrect local replacement. There is no quartic or kinetic counterterm in the bare potential. The number \(\mu_N\) generally diverges as \(N \to \infty\); only the running mass at each finite scale has a continuum limit. We never interchange these two limits of the mass. The explicit form of \(\mu_N\) is Theorem 22.2, and the limits of the coefficients are in Proposition 22.1.

Proof. Let \(D = \sup_q | \delta \mu_q |\) and \(X = \sup_{q, j} q^{3 / 2} v_1 (\delta \Theta^{[j]}_q)\). Both are finite at fixed \(N\): Lemma 20.7 gives \(a_2 (q) = O_N (q^{- 2})\), so the centred source is \(O_N (q^{- 3})\) and \(v_1 (\Theta_q) = O_N (q^{- 2})\). Put \(\rho_q = D / q + Xq^{- 2}\), which bounds \(| \delta \mu_q | / q + q^{- 1 / 2} v_1 (\delta \Theta_q)\). The trace comparison gives \(\mathcal{N}_{k, s} (\mathcal{T}_s [M^{[j], (1)}, h] -\mathcal{T}_s [M^{[j], (2)}, h]) \leqslant C \rho_s s^{1 / 2} r_s^{- k}\) at each level. Since the seed does not depend on the mass, integrating from \(q\) to \(\infty\) with \(k = 4\) and using the finite recursion,

Let \(R_0, \hat{R}_0\) be the two zero-field inverses. These are commuting Fourier multipliers, each with symbol bounded by \(C (q + c|k|^2)^{- 1}\) by (19.5), and \(\delta (R_0^2) = - (R_0^2 \hat{R}_0 + R_0 \hat{R}_0^2) \delta M\). So the local term of (20.8) varies by \(\lambda \text{Tr} ((R_0^2 \hat{R}_0 + R_0 \hat{R}_0^2) \delta M)\), to leading order \(2 \lambda \text{Tr} (R_0^3 \delta M)\). This is at most \(C \lambda (Dq^{- 3 / 2} + Xq^{- 5 / 2})\), as \(| \widehat{\delta \Theta} (k) | \leqslant 2 |k| v_1 (\delta \Theta)\) and \(\sum_k |k| (q + c|k|^2)^{- 3} \leqslant Cq^{- 1}\). Contracting the fourth vertex with \(R_0^2\), whose supremum is \(Cq^{- 1 / 2}\), and the vertex \(D^4 h^{[3]} (0)\) with \(\delta (R_0^2)\), whose supremum is at most \(C \rho_q q^{- 1 / 2}\) by the marked inverse bounds of Lemma 20.4, gives

The same bounds hold for the centred sources of the iteration at each level, whose kernels are integrated from infinity; as the kernels are outputs of the finite recursion, this gives directly \(X \leqslant C \lambda^2 D\), the bound at the first level feeding the second. Integrating the scalar part from \(q_{\ast}\) to \(q\),

For \(q_{\ast}\) large this gives \(D \leqslant 2 | \delta b_{\ast} |\), and integrating \(\delta \beta\) over \([q_{\ast}, \infty)\) gives the last bound.\(\Box\)

21Global coercivity and the complete large-field error

This section proves (12.2), Lemma 12.1 and Proposition 12.2 of Lecture 6.

21.1Clipping

Choose a smooth odd \(\tau\), equal to the identity on \([- 1, 1]\), constant outside \([- 2, 2]\), with \(0 \leqslant \tau' \leqslant 1\) and \(| \tau | < 2\). Define pointwise, as in (12.1),

The quartic and quadratic terms are not clipped. The finite chain rule gives globally, as in (12.2),

For the scale term use \(\partial_q \kappa_q y = (\kappa_q y - y \tau' (y / r_q)) / (2 q)\), of size \(Cr_q / q\), and differentiate the composition. These terms remain in the defect.

Proof. We verify the clipping budgets (21.2). In this proof \(h = h_q^{[3]}\) is the unclipped correction and \(J = D \kappa_q (y)\) is the diagonal operator multiplying by \(\tau' (y / r_q)\). Taylor on the unclipped ball gives \(\|Dh\|_{\infty} \leqslant Cg_q qr_q\) as well as the stated Hessian and higher budgets. In \(D^j (h \circ \kappa_q)\), each partition with \(m\) blocks has one \(D^m h\) and clipping factors whose product is bounded by \(Cr_q^{m - j}\). Indeed \(| \kappa_q^{(a)} | \leqslant C_a r_q^{1 - a}\) for \(a \geqslant 1\). Coincident slots give physical deltas integrating to one. Root any external slot and use symmetry of the \(h\) vertex to obtain the asserted Schur bound for \(2 \leqslant j \leqslant 4\).

For scale differentiation first use \(\partial_q h^{[3]} = \Pi \mathcal{T}_q [M^{[2]}, h^{[2]}]\). With \(\Gamma^{[2]} = U + h^{[2]}\) and \(H^{[2]} = (K_q + M^{[2]}_q + D^2 \Gamma^{[2]})^{- 1}\), differentiating this trace twice gives the two types

Lemma 20.1 bounds both rooted kernels by \(Ca_2 (q) qr_q^{- 2} \leqslant Cg_q\). Subtracting the quadratic jet at zero preserves this bound. Hence \(\textsf{S}_2 (\partial_q h^{[3]}) \leqslant Cg_q\) and \(\|D \partial_q h^{[3]} \|_{\infty} \leqslant Cg_q r_q\) on the ball. Write \(z = \kappa_q y\) and \(w = \partial_q \kappa_q y\). Globally,

Two derivatives of the first term cost \(Cg_q\) by the preceding two bounds. The second gives respectively \(D^3 h [J, J, w]\), \(D^2 h [\kappa_q'', w]\), the two terms \(D^2 h [J, Dw]\), and \(Dh D^2 w\). Each costs \(Cg_q\) in rooted Schur norm, using the displayed budgets. This proves the scale assertion in (21.2), including motion of the clipping scale.\(\Box\)

21.2Global coercivity

Write \(H_q (y) = (K_q + D^2 \bar{\Gamma}_q (y))^{- 1}\), as in (10.11).

Proof. Use (19.5) and \(\|D^2 \tilde{h} \|_{2 \to 2} \leqslant Cg_q q\) for the forms. The row Neumann series in \(H = (I + G (M + D^2 \tilde{h}))^{- 1} G\) has first-factor norm \(1 + C (g_q + g_q^2)\). Symmetry gives columns and interpolation gives \(p\). Every row of \(G\) has \(L^2\) norm at most \(\sqrt{a_2 (q)}\); Minkowski in the same series proves the covariance estimate. Finally use \(H^2 \leqslant (C / q) H\) and \(M_y HM_y \leqslant C \lambda^{- 1} I\). One must not square a Loewner inequality to obtain this assertion.\(\Box\)

21.3The complete defect

The defect is \(e_q (y) = \partial_q \bar{\Gamma}_q (y) - \frac{1}{2} \text{Tr} [H_q (y) - H_q (0)]\) and the drift is \(b_q (y) = - H_q (y) De_q (y)\), as in (10.11). Put \(\chi_q (y) = 1_{|y| > r_q}\), pointwise, and split the drift as

Proof. The exact gradient is

For this calculation let \(h = h_q^{[3]}\), \(z = \kappa_q y\), and \(J = D \kappa_q (y)\), the diagonal derivative of the clipping map, and write \(M' = \partial_q M_q\). Evenness, the zero scalar quadratic jet of \(e^{\text{o}}\) and its centred quadratic jet \(\mathcal{E}^c_q\) give \(\|De^{\text{o}} (z)\|_{\infty} \leqslant \textsf{S}_4 (e^{\text{o}}) (2 r_q)^3 / 6 + \|\mathcal{E}^c_q \|_{\text{row}} 2 r_q \leqslant C \lambda^{7 / 2} q^{- 3 / 2}\), by (20.12). Multiplication by \(J\) and the \(C / q\) inverse bound therefore give the bounded-field estimate in (21.4). Here are the identities for the large-field calculation. Put \(w = \partial_q \kappa_q y\), \(f_1 = Dh (z)\), \(f_2 = D^2 h (z)\), and \(f_3 = D^3 h (z)\). Here \(w\) is only the local abbreviation for the scale derivative of the clip. The unclipped and clipped inverses are

Their inverse difference, denoted \(\textsf{D}\) to distinguish it from field differentiation, obeys

The full gradient difference is

These are the groups of Section 12.3 of Lecture 6. Here \(J\) in the last line acts on the open slot, \(\kappa_q''\) denotes the pointwise second derivative of the scalar clipping function, and products such as \(\kappa_q'' uv\) are pointwise. In the next identity the symbol \(\sum_{\text{cyc}}\) denotes the three cyclic permutations of the directions \(u, v, v'\). Retain the entire chain rule

Suppress \(q\) and put \(r = r_q\), \(g = g_q\), \(a_2 = a_2 (q)\), \(\chi = \chi_q (y)\) and \(t = y \chi\). Let

Clipping obeys the following pointwise bounds, all uniform in \(y\):

The estimates for \(H\) hold also for \(H^{\text{o}}\) on the unclipped ball, by the same Neumann-series proof. In particular, for \(H^a\) either of the inverses \(H, H^{\text{o}}\),

Receive every inverse insertion before estimating the field. Write \(\mathcal{D}_2 = H^2 - (H^{\text{o}})^2\). In each term of its exact identity the two outer inverse chains have product of \(\ell\) norms at most \(Ca_2 / q\). For a local insertion use Lemma 20.3. In an insertion \((J - I) f_2 J\) or \(f_2 (J - I)\), absorb \(f_2, J\) into the adjacent chain; this costs at most \(\sigma_2\) in its \(\ell\) norm. Finally \(|f_1 \kappa'' | \leqslant C \sigma_1 \chi / r\). The lemma therefore gives, for all \(p\),

In the last inequality use \(\sigma_2 + \sigma_1 / r \leqslant Cgq\), \(\chi \leqslant \lambda y^2 \chi / q\), and small \(g\). This argument controls the column mass of the nonlocal contraction as well as its row mass.

Receive the third-order chain rule. In its first term, after removing the open-slot \(J\), the difference of the two contracted-slot factors is bounded by \(\chi (u) + \chi (v)\). The last assertion of Lemma 20.3, with \(\sup_{x, u} |H^2 (x, u) | \leqslant Ca_2\), gives \(Ca_2 \sigma_3 \| \chi \|_p\). Of the three second-derivative terms, one identifies the two contracted slots and the other two identify the open slot with a contracted slot. Their majorant kernels have Schur masses at most \(Ca_2 \sigma_2 / r\). The last term identifies all three slots and costs \(Ca_2 \sigma_1 / r^2\). Write \(\mathcal{E}_3 = D^3 (h \circ \kappa_q) - Jf_3\). Consequently

Here \(q / r^2 = \lambda\) and \(\lambda a_2 \leqslant Cg\). This is an operator estimate; the marked input sites can produce output at other spatial sites.

Bound the six groups in the complete gradient difference. The following ledger follows directly from (21.7)–(21.9):

For the first row split the expression exactly as \(M' (y - z) + (I - J) M' z\). The first part uses \(m (M') \leqslant Cg\); the second uses \(\|M' z\|_{\infty} \leqslant Cgr\) and \(\chi \leqslant |t| / r\). The second row uses the \(w, Dw\) bounds and \(\sigma_1, \sigma_2\). For the fourth row note \(\lambda^2 ra_2 / q = \lambda a_2 / r \leqslant Cg / r\); for the sixth use \(\sigma_3 \lambda a_2 / q \leqslant Cg^2 / r\).

Put \(\mathfrak{D}_q (y) = De_q (y) - D \kappa_q (y) De_q^{\text{o}} (\kappa_q y)\). Summing the ledger gives, before multiplication by \(H\), for all \(y\) and \(p\),

This is the bound used in Lemma 18.1 of Lecture 7; rows one, two, three–four and five–six are the four groups of Section 12.3 of Lecture 6. Since \(b^{\text{l}} = - H\mathfrak{D}\), using \(\|H\|_{p \to p} \leqslant C / q\) proves

On the tail \(|t| + r^{- 1} y^2 \chi \leqslant 2 r^{- 2} |t|^3\); squaring the \(p = 2\) inequality gives \(\|b^{\text{l}} \|_2^2 \leqslant C \lambda^4 q^{- 5} \|t\|_6^6\). This retains the same coefficients needed for the backward energy.

For the global linear estimate, freeze \(H (y)^2\) in (21.6) and write \(D^3 \tilde{h} (y) = \int_0^1 D^4 \tilde{h} (uy) [y] \hspace{0.17em} \mathrm{d} u\). Collapsing the covariance slots gives a two-slot kernel with both Schur masses at most \(Ca_2 s (D^4 \tilde{h}) \leqslant Ca_2 gq / r^2 \leqslant Cg^2\). The other three terms in that gradient have \(L^p\) bounds \(Cg \|y\|_p\), using \(m (M') \leqslant Cg\), \(s (D^2 \partial_q \tilde{h}) \leqslant Cg\), and \(\lambda \| \text{diag} H^2 \|_{\infty} \leqslant Cg\). Multiplication by \(H\) proves \(\|b (y)\|_p \leqslant Cgq^{- 1} \|y\|_p\). In particular, at \(q = q_{\ast}\), the last two terms of (21.6), which form \(D \Upsilon\) in (12.8) of Section 12.4 of Lecture 6, are bounded by \(Cg_{q_{\ast}} \|y\|_p\). Finally \(D \varrho = C_q M + C_q D^2 \tilde{h}\). The centred convolution estimate bounds the first summand by \(C (| \mu | / q + \eta_q (\Theta))\) on every \(L^p\); the Schur bounds for \(C_q\) and the clipped Hessian bound the second by \(Cg\). This proves (21.5). As \(\varrho_q (0) = 0\) by evenness and \(DU = \frac{\lambda}{3} y^3\), the equation \(K_q y + D \bar{\Gamma}_q (y) = K_q x\) is (12.5) of Lecture 6.\(\Box\)

22The bare mass and its limit

This section gives the consistency of the mass path with a moving boundary value and the proof of the explicit counterterm, Theorem 11.3 of Lecture 6. All constants may depend on \(\lambda, q_0, B_0\), but not on \(N\) or on the boundary value.

Proof. This is Theorem 28.3, whose proof does not use this section.\(\Box\)

Proof. Expansion. Expand \(R_0 = C - CMC + CMCMR_0\) with \(C = C_q\). Then \(- \lambda \text{Tr} R_0^2 = - \lambda a_2 + 2 \lambda \text{Tr} (C^3 M) + O (\lambda a_2 \rho^2)\), where \(\rho = | \mu | / q + \eta_q \leqslant C g_q\). As \(a_2' = - 2 \text{Tr} C^3\), this is \(- \lambda a_2 - \lambda \mu a_2' + 2 \lambda \text{Tr} (C^3 \Theta)\) up to the error, and \(| \text{Tr} (C^3 \Theta) | \leqslant Cq^{- 1} v_1 (\Theta) \leqslant C \lambda^2 q^{- 3 / 2}\). For the vertex, write \(h^{[3]} =\mathcal{L}+ (h^{[2]} -\mathcal{L}) + (h^{[3]} - h^{[2]})\). The increments have fourth rooted norm at most \(Cg_q^2 \lambda\), and contracted with \(R_0^2\), whose supremum is \(Cq^{- 1 / 2}\), they give \(O (\lambda^3 q^{- 3 / 2})\). The quartic part of the seed is \(L (y) = - \frac{\lambda^2}{4} \int \int C (x, z)^2 y (x)^2 y (z)^2\). So

where \(\cdot\) denotes the pointwise product of kernels. Its constant-mode eigenvalue is \(\lambda^2 a_2 \text{diag} (R_0^2) + 2 \lambda^2 \int C^2 R_0^2\), and replacing \(R_0^2\) by \(C^{(2)}\) costs a relative factor \(Cg_q\), that is, \(O (\lambda^3 q^{- 3 / 2})\). This proves the expansion.

Integration. Three identities: \(\partial_q \text{Tr} C_q = - a_2\); \(\partial_q C_q (x) = - C_q^{(2)} (x)\), so \(\partial_q \int C_q^3 = - 3 \int C_q^2 C_q^{(2)}\); and \(\partial_q (\mu a_2) = \beta a_2 + \mu a_2'\), so \(- \lambda \mu a_2' + \lambda^2 a_2^2 = - \lambda \partial_q (\mu a_2) + \lambda (\beta + \lambda a_2) a_2\). At fixed \(N\), as \(q \to \infty\), \(\text{Tr} C_q\), \(\int C_q^3\) and \(\mu_q a_2 (q)\) tend to \(0\). Since \(\lambda^2 \cdot 2 \cdot \frac{1}{3} = \frac{2}{3} \lambda^2\), integrating \(\beta\) over \([q_{\ast}, \infty)\) gives \(\mu_N = b_{\ast} (1 + \lambda a_2 (q_{\ast})) - \lambda \text{Tr} C_{q_{\ast}} + \frac{2}{3} \lambda^2 \int C_{q_{\ast}}^3 + \int (\mathcal{R}+ \lambda (\beta + \lambda a_2) a_2)\), which is (22.1). The last integrand is \(O (\lambda^3 q^{- 3 / 2})\), because \(\beta + \lambda a_2 = O (\lambda^2 / q)\): indeed \(\mu a_2' = O (\lambda q^{1 / 2} q^{- 3 / 2})\).

Limit. The multiplier of \(\text{Tr} C - \text{Tr} C_{q_{\ast}}\) is \((q_{\ast} - q_0) ((q_0 + c_k) (q_{\ast} + c_k))^{- 1} \lesssim (1 + |k|)^{- 4}\), summable, with \(c_k \geqslant c |k|^2\), so this sum converges. Next \(C^3 - C_{q_{\ast}}^3 = (C - C_{q_{\ast}}) (C^2 + CC_{q_{\ast}} + C_{q_{\ast}}^2)\), where \(C - C_{q_{\ast}}\) converges uniformly, by the same summability, and the second factor converges in \(L^1\). At fixed \(q\), \(\beta_q^N\), \(\mu_q^N\) and the explicit terms converge by Proposition 22.1 and the convergence of the free kernels, hence so does \(\mathcal{R}_q^N\). The integrand is dominated by \(C \lambda^3 q^{- 3 / 2}\), so \(f_N \to f\) pointwise.

Inversion. By Proposition 20.8 and \(\lambda a_2 (q_{\ast}) \leqslant Cg_{q_{\ast}}\), \(f_N\) is increasing with \(|f_N (b_{\ast, 1}) - f_N (b_{\ast, 2}) - \delta b_{\ast} | \leqslant Cg_{q_{\ast}} | \delta b_{\ast} |\), and the same holds for \(f\). By (22.1), \(f_N (0) = \lambda (\text{Tr} C - \text{Tr} C_{q_{\ast}}) + O (\log q_{\ast})\), and \(\text{Tr} C - \text{Tr} C_{q_{\ast}} \leqslant C (q_0) + q_{\ast}^{1 / 2} / 10\) uniformly in \(N\) (the continuum value of the coefficient is \(\frac{1}{4 \pi} < \frac{1}{10}\), and the lattice sum is bounded by \(\frac{1}{10}\) uniformly in \(N\)). So for \(B_0\) above a universal constant and then \(q_{\ast}\) large depending on \(\lambda, q_0, B_0, r\), \(f_N (\mathcal{B})\) contains \(r\) for all \(N\). Monotonicity and pointwise convergence give \(b_{\ast, N} \to f^{- 1} (r)\).\(\Box\)

The measure depends only on \(\mu_N\), so the choice of \(q_{\ast}\) is immaterial for (22.2): we may take any large \(q_{\ast}\).

23Identifying the deterministic coefficients: conventions

This part proves the statements of Section 18.1 of Lecture 7 and Lemma 18.3: the convergence of the coefficients along fields, which enter Theorems 18.4 and 18.6. The estimates of Proposition 11.2 do not by themselves identify the coefficients of a limit: a weakly converging vertex cannot automatically be contracted against a varying covariance. That additional assertion is proved here.

Notation. We use that of Lectures 6–7: \(K_q = q - \Delta_N\), \(C_q = K_q^{- 1}\), \(g_q = \lambda q^{- 1 / 2}\), \(r_q = (q / \lambda)^{1 / 2}\), \(a_2 (q) = \text{Tr} C_q^2\), the quadratic path \(M_q = \mu_q I + \Theta_q\), \(R_{0, q} = (K_q + M_q)^{- 1}\), \(G_q (y) = (K_q + \lambda y^2)^{- 1}\), the seed \(\mathcal{L}_q\), the corrections \(h^{[j]}_q\) of (11.8), the sources \(\mathfrak{S}_q [M, h]\) of (20.8), with \(\mathfrak{S}_q 1 = \beta_q 1\) at the final level and centred parts \(\mathfrak{S}_q^{\circ}\), the centred kernels \(\Theta^{[j]}_q\) with \(\Theta_q = \Theta^{[3]}_q\), the clip \(\kappa_q\), \(\tilde{h}_q = h^{[3]}_q \circ \kappa_q\), the action \(\bar{\Gamma}_q\), the response \(H_q (y) = (K_q + D^2 \bar{\Gamma}_q (y))^{- 1}\), the defect \(e_q\), the drift \(b_q = - H_q De_q\), and the coordinate correction \(\varrho_q (y) = C_q (M_q y + D \tilde{h}_q (y))\) of (12.5). \(\text{Ctr}_R T\) contracts two slots of the tensor \(T\) with the kernel \(R\). The letter \(H\) without a scale index is the space \(L^2 (\mathbb{T}^3)\); the response always carries a scale index or a field. \(\text{HS}\) is the Hilbert–Schmidt class on \(H\). The boundary value (below) is \(b_{\ast}\); \(b_q (y)\) is the drift.

Superscripts or subscripts \(N\) distinguish the cutoffs: \(\mu^N = (\mu^N_q)_{q \geqslant q_{\ast}}\) is a running mass path, while \(\mu_N\) is reserved for the single bare mass. \(k \in \mathbb{Z}^3\) and \(e_k (x) = \mathrm{e}^{2 \pi ik \cdot x}\) (the \(k\) of Lecture 6 is \(2 \pi\) times this one); \(e^N_k\) is the restriction of \(e_k\) to \(\Lambda_N\), and \(\Lambda^{\ast}_N\) is the centred cube of \(N^3\) frequencies. On \(\Lambda^{\ast}_N\) the symbol \(c_k = 4 N^2 \sum_i \text{sin}^2 (\pi k_i / N)\) of \(- \Delta_N\) satisfies \(c |k|^2 \leqslant c_k \leqslant 4 \pi^2 |k|^2\).

Identifications. Fields are embedded by the cell function \(\iota_N\), operators by \(\iota_N \cdot j_N\), with \(j_N = \iota_N^{\ast}\), \(j_N \iota_N = I\) and \(\textsf{P}_N = \iota_N j_N \to I\) strongly. All embedded inverses are \(\iota_N Rj_N\) on \(H\), with zero action on the orthogonal complement of the cell functions. Their resolvent identities are valid in this convention; an inverse on the complementary subspace is never required. For a smooth \(\phi\) we put \(\phi_N = j_N \phi\), so \(\| \phi_N \|_{\infty} \leqslant \| \phi \|_{\infty}\) and \(\| \iota_N \phi_N - \phi \|_{\infty} \leqslant C \| \phi \|_{C^1} N^{- 1}\).

Boundary value. Fix \(B_0 < \infty\), let \(\mathcal{B}= [- B_0 \lambda q_{\ast}^{1 / 2}, B_0 \lambda q_{\ast}^{1 / 2}]\), fix \(b_{\ast, N} \to b_{\ast}\) in \(\mathcal{B}\), and take \(q_{\ast}\) large depending on \(\lambda, B_0\). Throughout, the lattice coefficients are those of the system (11.8)–(11.9) with \(\mu^N_{q_{\ast}} = b_{\ast, N}\), the centred kernels integrated from infinity; by Proposition 20.6 the level masses \(M = \mu I + \Theta^{[j]}\) lie in the ball \(\|M\|_{\ast} \leqslant R = B_0 + C_{\ast}\), where

Constants may depend on \(\lambda, B_0\) and on the compact scale interval \(I \subset [q_{\ast}, \infty)\) when so indicated, and never on \(N\). The bounds of Proposition 11.2, of (12.2), of Lemma 12.1 and of Proposition 12.2 hold in this ball.

Measures. We use signed Radon measures, that is, signed measures with finite variation on compact sets. For such a \(\sigma\), \(| \sigma |\) is its total variation measure and \(\| \sigma \|_{\text{TV}} = | \sigma | (E)\) on its underlying space \(E\). \(\delta_z\) has unit mass at \(z\). Weak convergence on a compact space means convergence against every continuous test; vague convergence off zero uses continuous tests with compact support in the punctured torus. An annular test is supported a positive distance from zero.

24The topology of the centred quadratic kernel

Let \(\mathcal{Q}_1\) be the space of even signed Radon measures on \(\mathbb{T}^3 \setminus 0\) with norm

Such a measure acts by the compensated formula

For a lattice kernel the off-diagonal measure assigns mass \(N^{- 3} \Theta_N (0, z)\) to \(z \neq 0\); its diagonal is supplied by centring, \(\Theta_N 1 = 0\). There is no independent diagonal mass in \(\mathcal{Q}_1\): that coefficient belongs to \(\mu I\). The multiplier \(\hat{\Theta} (k)\) is characterised by \(\Theta e_k = \hat{\Theta} (k) e_k\).

Proof. Multiplication by \(d (0, z)\) identifies \(\mathcal{Q}_1\) isometrically with the even finite signed measures on the punctured torus, a Banach space in total variation. On each closed annulus the weighted variation is the supremum over continuous tests dominated by \(d\). Taking increasing annuli gives lower semicontinuity.

For \(d (0, z) < \delta\) the second difference of a smooth function satisfies

The small displacement contribution is therefore uniformly negligible, as the first moments are bounded. Away from \(d (0, z) < \delta\) use a continuous cutoff and the vague convergence; a finite net in \(x\) and the same bound for derivatives give uniformity in \(x\). Cell averages commute with translations by lattice displacements, so replacing \(f\) by its cell averages gives the same limit. There is no large distance tail on the torus.

For a Fourier character the multiplier is \(\int (\text{cos} (2 \pi k \cdot z) - 1) \Theta (\hspace{0.17em} \mathrm{d} z)\), and \(| \text{cos} (2 \pi k \cdot z) - 1| \leqslant C |k| d (0, z)\) proves the bound. This also justifies a Fourier approximation of a smooth \(f\), uniformly over the moment bounded family. The function \(\text{cos} (2 \pi k \cdot z) - 1\) vanishes to second order at \(z = 0\), so the first part with \(f = e_k\) gives the convergence of the multipliers. The scale uniform claims follow by the identical estimates. Bounded first moments prevent a hidden second order differential operator from surviving at zero.\(\Box\)

The strong metric \(d_{\ast}\) of (23.1), restricted to scalar paths, is the one used for the fixed point; on the centred kernels it only records their bounds. It is different from the weak path topology: lattice measures need not converge in total variation to their continuum limits. Both lattice and continuum paths are defined on \([q_{\ast}, \infty)\), so the same metric is used for all their contraction estimates. Weak convergence is still only local in scale.

25Physical vertex measures and their derivatives

If \(D^k f_N (y)\) has physical tensor kernel \(T_N\), associate the atomic measure

Integrating smooth products against it evaluates the derivative on their lattice samples. Evaluation on cell averages differs by \(o (1)\), since samples and averages of a fixed smooth function differ uniformly by \(O (N^{- 1})\) and the tensor variation is bounded. We use atomic measures for weak convergence and cell kernels for operator comparisons. As in Section 20.1, \(s (T)\) is the rooted norm.

Proof. The total variation is at most \(C\), since the root variable has volume one. Weak compactness of bounded measures gives a subsequence. For a continuous nonnegative root test \(a\) and a continuous product space test \(\psi\) with \(| \psi | \leqslant 1\), the rooted estimate gives

Pass to the limit and take the supremum over \(\psi\) to obtain the marginal variation bound. Insert a distance in the tests for the moment version. A symmetric two-slot measure \(\textsf{T}\) therefore obeys

This defines its bounded operator, first on smooth functions, then by density on \(H\).\(\Box\)

Proof. Restrict to \(\phi + \sum_{i = 1}^d t_i u_i\), with smooth directions \(u_i\) and parameters a positive distance inside the domain. A mixed finite difference of order \(k \leqslant m\) and step \(\delta\) approximates the corresponding derivative with error \(C \delta\), by the bound on the derivative of order \(k + 1\). At fixed \(\delta\) its finitely many scalar values converge. Let \(N \to \infty\), then \(\delta \to 0\). The next derivative bound and finite parameter nets give uniformity on compact parameter sets.

Products of smooth one-variable tests determine a measure on the product torus: finite Fourier sums approximate every continuous product space test uniformly, and the common variation bound removes that approximation. This identifies the full weak vertex limit, not just a selected collection of tensor contractions. The limiting scalar action on these families has the stated derivatives, since the fundamental theorem of calculus passes to the limit on segments.\(\Box\)

For even corrections with zero quadratic jet, Taylor's formula also gives

Thus the Hessian Schur bounds follow from the bounds on the fourth derivative on every bounded field set. Local diagonal terms cause no difficulty: they are diagonal measures, not ordinary functions of two independent variables.

26Compact resolvents receive weak Hessians

The four lemmas of this section (compact sandwich, and contraction against a moving covariance) are the second and third tools listed in Section 18.1 of Lecture 7.

Proof. For each fixed Fourier mode the multipliers converge, by Lemma 24.1 and the convergence of the lattice Laplacian symbol. Their absolute values are bounded by \(C (q + c|k|^2)^{- 1}\), by the small relative form bound: \(| \mu_q | \leqslant Rg_q q\) and \(| \hat{\Theta}_q (k) | \leqslant CRg_q \lambda |k| \leqslant CRg_q (g_q^2 q + |k|^2)\). The squared tail beyond \(|k| = L\) is therefore at most \(C_I \sum_{|k| > L} (1 + |k|^2)^{- 2} \leqslant C_I / L\). The embedded lattice Fourier characters converge uniformly to their continuum characters on each fixed finite set of modes. The low and high frequency split proves Hilbert–Schmidt convergence.

For the multiplication potential put \(V_N = \lambda (\iota_N j_N \phi)^2\) and \(V = \lambda \phi^2\), so \(\|V_N - V\|_{\infty} \to 0\). If \(A_N, A\) denote the preceding background inverses and \(G_N, G\) the inverses with \(V_N, V\), the exact identity is

All outer factors have bounded operator norms on the fixed profile, and the middle difference is Hilbert–Schmidt. The identity follows by expanding and using \(G_N = (I - G_N V_N) A_N\) and \(G = A (I - VG)\). This proves the assertion without an expansion in the field amplitude.\(\Box\)

Proof. A uniformly bounded weakly convergent sequence \(E_N \to 0\) satisfies \(AE_N D \to 0\) in Hilbert–Schmidt norm whenever \(A, D\) are Hilbert–Schmidt. Indeed, approximate both outer factors by finite rank operators: the finite matrix elements converge, and the two errors are bounded using \(\|XYZ\|_{\text{HS}} \leqslant \|X\|_{\text{HS}} \|Y\| \|Z\|\). Apply this to \(G (\textsf{B}_N - \textsf{B}) R\). Also \((G_N - G) (\textsf{B}_N - \textsf{B}) R \to 0\) in Hilbert–Schmidt norm. Now use

where \(R_N = (I + G_N \textsf{B}_N)^{- 1} G_N\) in the last term. The inverse factors are uniformly bounded by \((1 - \theta)^{- 1}\), so both terms vanish. Weak measure convergence gives weak operator convergence on smooth product tests, and the Schur bound extends it to arbitrary \(L^2\) tests by density.\(\Box\)

In the applications \(G_N\) includes the quadratic background and the nonnegative local potential. Here \(\phi_N = j_N \phi\) for a fixed smooth bounded profile \(\phi\), \(h_N\) is the nonlinear correction at whichever finite depth is being considered, and \(\textsf{B}_N = D^2 h_N (\phi_N)\). The finite hierarchy gives \(\| \textsf{B}_N \| \leqslant Cg_q q\), and \(\|G_N \| \leqslant C / q\), so the same large \(q_{\ast}\) ensures the smallness required in the lemma. The limit inherits these bounds and defines a positive inverse before the convergence argument is used.

Proof. For \(R_{0, N} = (K_q + M_q)^{- 1}\), Parseval and the multiplier bound give

The sum converges because \(a < \frac{1}{2}\). At a bounded field, write \(R_N = R_{0, N} - R_{0, N} (\lambda \phi_N^2 + \textsf{B}_N) R_N\); the factor to the right of \(R_{0, N}\) is bounded on \(L^2\). Taking row differences proves (26.1). Since \(R_N^2 (x, z)\) is the inner product of the two inverse rows, their common \(L^2\) bound proves the covariance modulus.

For cell embeddings that modulus has an additional error \(CN^{- a}\), as each point is replaced by its cell centre. The kernels are precompact in the uniform norm up to this vanishing error, for instance by interpolating their grid values continuously. On the other hand Hilbert–Schmidt convergence implies \(R_N^2 \to R^2\) in Hilbert–Schmidt norm, by the bounded operator norms. Thus every uniform subsequential limit equals the same \(L^2\) kernel. This proves uniform convergence and continuity.\(\Box\)

Proof. After testing the remaining slots by a continuous function, split the difference as

The first term tends to zero by uniform convergence and bounded variation; the second by weak convergence against the fixed continuous kernel. The use of atomic vertices and cell covariances is consistent: the former evaluate the latter at their grid values, and the uniform cell error tends to zero. For a Hessian the trace identity follows first for finite rank approximations of the smoothing inverses, then by the same kernel bound. Equivalently, the Schur factorisation of its symmetric measure as \(M_{\sqrt{a}} \textsf{E} M_{\sqrt{a}}\), \(\| \textsf{E} \| \leqslant 1\), makes the double sandwich trace class; here \(a (x) \hspace{0.17em} \mathrm{d} x\) is its variation marginal. Thus no trace of a single, unsquared divergent inverse is taken.\(\Box\)

27Convergence of the seed and the two trace steps

Proof. Subtract before taking a diagonal. The exact force identity is

It follows from \(D\mathcal{L}= \lambda \phi \text{diag} (G - C)\) and \(G - C = - C \lambda \phi^2 G\). Both inverses converge in Hilbert–Schmidt norm by Lemma 26.1. Cauchy–Schwarz in their two kernel variables proves \(L^1\) convergence of this force. Its uniform bound \(C_I \| \phi \|_{\infty}^3\) upgrades convergence to every finite spatial \(L^p\). Integrate \(\langle D\mathcal{L}_q (t \phi), \phi \rangle\) over \(t \in [0, 1]\) to identify the scalar action, with value zero at the origin. The ten derivative bounds of Proposition 11.2 and Lemma 25.2 give the nine tested derivatives. All estimates are uniform on compact profile families and scale intervals. In particular neither \(\text{diag} C_q\) nor \(\text{diag} G_q\) is assigned a separate continuum limit.\(\Box\)

Proof. The assertion at depth one is Lemma 27.1. Suppose scalar and tested convergence hold at depth \(j\). The Hessians converge weakly with uniform Schur bounds. Lemmas 26.1 and 26.2 identify the inverses with Hessian \(\theta D^2 h^{[j]}\), \(0 \leqslant \theta \leqslant 1\). The convergence is uniform in \(\theta\), since the resolvent identity gives a common Lipschitz bound in that parameter.

Let \(G_M = (K_q + M_q + \lambda \phi^2)^{- 1}\) and \(R_{\theta h} = (K_q + M_q + \lambda \phi^2 + \theta D^2 h^{[j]} (\phi))^{- 1}\). The scalar trace map decomposes exactly as

The first term converges as a product of Hilbert–Schmidt operators. The second converges by Lemma 26.4. Thus the scalar trace map converges at every admissible smooth profile. The finite derivative bounds and finite differences also identify its quadratic jet at zero, so its projection by \(\Pi\) converges.

It remains to pass the improper scale integral in (11.8). A fixed profile admissible at scale \(q\) remains admissible at all later scales. The initial trace comparison (Lemma 20.4), the common coefficient ball, and Taylor's formula to fourth order give

Consequently the omitted tail beyond \(L\) is at most \(C \lambda^3 L^{- 1} \| \phi \|_4^4\), uniformly in the cutoffs. Pass to the limit first on \([q, L]\), then send \(L \to \infty\). This proves scalar convergence at depth \(j + 1\). Its derivative budget is eight at depth two and six at depth three, so finite differences identify derivatives through seven and five. These orders include the fourth zero-field vertex needed by the mass equation. There is one genuinely unused derivative at the final step; it cannot be omitted.

The identities \(\partial_q h^{[j + 1]} = \Pi \mathcal{T}_q [M^{[j]}, h^{[j]}]\) and the corresponding differentiated estimates give local scale equicontinuity. Compact parameter nets give local uniformity. Only the two stated steps are used; no infinite derivative hierarchy or infinite field Taylor series has been introduced.\(\Box\)

28The scalar fixed point converges

The continuum source is a measure valued kernel. Weak convergence does not supply an integral in the total variation norm. The following construction is sufficient and avoids that assumption.

Proof. Write \(d (z) = d (0, z)\). On the punctured torus the functional

is bounded on \(C_0 (\mathbb{T}^3 \setminus 0)\) with the uniform norm: the space of continuous \(\psi\) such that the set where \(| \psi | \geqslant \varepsilon\) is compact in the punctured torus for every \(\varepsilon > 0\). The norm of the functional is at most \(\int_q^{\infty} f\). The inner integrals are measurable: first take compactly supported tests and then approximate in the uniform norm, using the bound on \(d |\mathfrak{C}_s |\). The Riesz representation theorem gives a finite signed measure \(\sigma_q\) for this functional. Define \(\Theta_q = d^{- 1} \sigma_q\) off zero. It is a locally finite signed Radon measure with \(\int d | \Theta_q | \leqslant \int_q^{\infty} f\) and the required test integrals. Uniqueness follows from compactly supported tests. The identical argument on \([q, t]\) proves the difference bound and continuity. For cutoff convergence, first integrate on \([q, L]\) and then bound the common tail by \(\int_L^{\infty} f\) times the weighted test norm. This is uniform for \(q\) in a compact interval. Evenness passes through the test prescription.

For the final assertion, with \(\delta_z^2 u = u (\cdot + z) + u (\cdot - z) - 2 u\), centring and evenness give on smooth \(u\)

where \(\langle \mathfrak{S}_s \rangle =\mathfrak{S}_s (\mathbb{T}^3)\); the scalar \(\beta_s\) enters only the scalar part. This integral is legitimate even for an infinite first moment measure: \(\| \delta_z^2 u\|_2 \leqslant C_u d (z)^2\). The preceding weak integral and Fubini, with the integrable first moment bound, give the first identity in (28.2), initially against smooth tests. The constant \(b_{\ast}\) cancels in the difference. To justify the strong \(H\) form, convolution by \(\mathfrak{S}_s\) has norm at most \(\|\mathfrak{S}_s \|_{\text{TV}}\). For a finite Fourier sum, \(\mathfrak{S}_s u\) is continuous in \(H\) by weak continuity of \(\mathfrak{S}_s\). Approximation by Fourier sums and the locally uniform total variation bound give this continuity for every \(u \in H\). Thus the right side is the integral of a continuous \(H\)-valued function, and its fundamental theorem of calculus proves the derivative assertion. On each compact scale interval \(M_t - M_q\) consequently extends to a bounded operator on \(H\), even when the individual \(M_q\) are only defined on a dense domain. No Banach space differentiability theorem for measures is used.\(\Box\)

Remark 28.2. (Norm continuity does not imply a measure derivative) Let \(e_1 = (1, 0, 0)\) and, for \(0 < a < t < c < \frac{1}{4}\), set \(\mathfrak{C}_t = \frac{1}{2} (\delta_{te_1} + \delta_{- te_1})\) off zero. Let \(F_t = \int_a^t \mathfrak{C}_s \hspace{0.17em} \mathrm{d} s\) in the weak sense, with diagonal supplied by centring. Then \(F\) is Lipschitz in \(\mathcal{Q}_1\), but

Indeed the difference quotient is distributed along two short arcs, whereas \(\mathfrak{C}_t\) consists of their endpoint atoms. They are mutually singular, so the first moments add. The quotient converges weakly to \(\mathfrak{C}_t\), so there is no derivative in the \(v_1\) norm. On a Fourier mode the centred multiplier is instead \(\text{cos} (2 \pi k_1 t) - 1\), and differentiating the integrated multiplier is valid. This illustrates exactly the distinction in the lemma.

Proof. Step 0: reduction to a fixed boundary value. For fixed \(N\) and \(b_{\ast}' \in \mathcal{B}\) write \(\mu^{N, b_{\ast}'}\) for the scalar path with boundary value \(b_{\ast}'\), and \(M^{N, b_{\ast}'}\) for any of its level masses. By Proposition 20.8, uniformly in \(N\),

Thus it is enough to prove the theorem for the fixed value \(b_{\ast}\): the tests below are Lipschitz in \(d_{\ast}\), and the assertions on the corrections follow from Proposition 27.2, which only requires weak path convergence of its input. We write \(\mathcal{A}_N\) for the scalar map defined by (11.9) with boundary value \(b_{\ast}\), \(\mathcal{A}_N (\mu) = b_{\ast} + \int_{q_{\ast}}^{\cdot} \beta_s [\mu] \hspace{0.17em} \mathrm{d} s\).

Step 1: the map on continuum paths. The continuum map must be defined on its whole ball, not just on a particular extracted sequence. Its argument is a scalar path, which is its own lattice approximation; the centred kernels are not arguments but outputs, constructed level by level in Step 2. Proposition 27.2 and Step 2 therefore define the continuum recursion and \(\mathcal{A} (\mu)\) for every continuous scalar path in the ball, with no approximation of measure-valued inputs. Apply the finite cutoff contraction estimate of Proposition 20.6 and pass to the limit: after exhausting scale intervals this gives the same contraction estimate for \(\mathcal{A}\).

Step 2: one application of the map. Let \(\mu^N\) be scalar paths in the ball converging locally uniformly to \(\mu\). Proceed level by level, starting from \(\Theta^{[1], N} = 0\) and the seed, which converges by Lemma 27.1. Suppose the level masses \(M^{[j], N}\) converge in the weak path topology to \(M^{[j]}\), and \(h^{[j], N}\) as in Proposition 27.2, which identifies the fourth zero-field vertex. Its contraction with \(R_0^2\) converges by Lemma 26.4; the local contribution \(- \lambda \text{Tr} R_0^2\) converges by Hilbert–Schmidt convergence. It follows that \(\mathfrak{S}_q [M^{[j], N}, h^{[j], N}]\) converges weakly as a two-slot measure, locally uniformly in scale. The limit is invariant under every torus shift: approximate that shift by lattice shifts in each continuous test. It is therefore also a convolution measure on the relative spatial variable. Its uniform row bound gives the limiting total variation bound \(\|\mathfrak{S}_q \|_{\text{TV}} \leqslant C (g_q + g_q^2)\). The uniform weak convergence and finite cutoff continuity give weak continuity of this source.

For the centred integral toward infinity, the bound of Proposition 20.6 is

Its upper tail in the first moment norm is at most \(2 C \lambda^2 L^{- 1 / 2}\). Lemma 28.1 constructs this integral without a strong measurability assumption in the measure norm. It proves convergence against annular tests; the small displacement estimate of Lemma 24.1 gives compensated tests, uniformly on compact scale intervals. So \(\Theta^{[j + 1], N} \to \Theta^{[j + 1]}\) in the weak path topology, and Proposition 27.2 gives the next primitive. After the two levels, testing both external slots of the final source by the constant function one identifies \(\beta^N_q\), since the volume is one. Hence the scalar integral from \(q_{\ast}\) to a fixed \(q\) converges, and the value \(b_{\ast}\) at \(q_{\ast}\) passes through unchanged. Lemma 28.1 gives local Lipschitz continuity of the centred kernels in the first moment norm; the scalar source gives it for \(\mu\). The identity \(M' = \beta I +\mathfrak{S}_q [M^{[2]}, h^{[2]}]^{\circ}\) in the continuum has the strong vector meaning (28.2). In particular the term \(M' y\) in the complete gradient (29.1) below means a bounded convolution, also on nonsmooth \(H\) inputs. No derivative in total variation norm is needed to define the drift. Lower semicontinuity keeps the limits in the same balls. There is no infinite upper scale integral for the scalar mass.

Step 3: iteration. Iterate \(\mathcal{A}_N\) from the constant path \(\mu \equiv b_{\ast}\), which lies in the ball as \(b_{\ast} \in \mathcal{B}\), and denote the \(r\)th iterate by \(\mu_N^{(r)}\). Write \(\mu_N = \mu^{N, b_{\ast}}\). For each fixed \(r\), induction using Step 2 proves convergence to a continuum iterate \(\mu^{(r)}\). The contraction estimate has a single \(\theta < 1\), independent of \(N\), and gives

These inequalities pass to the limit on compact scale intervals followed by exhaustion of the half-line. The weighted space of continuous scalar paths is complete. Therefore \(\mu^{(r)}\) tends to a path \(\mu\), with error \(C \theta^r\), and \(\mu_N \to \mu\) locally uniformly. The continuum map is Lipschitz by the difference estimates, so \(\mu =\mathcal{A} (\mu)\) and the contraction proves uniqueness in the ball. Step 2 with input \(\mu_N\) then gives the convergence of the centred kernels, of the corrections and of the sources, and the bounds \(v_1 (\Theta^{[j]}_q) \leqslant C \lambda^2 q^{- 1 / 2}\) pass to the limit.

Step 4: conclusion along the moving boundary. By (28.4), the scalar paths and the level masses with boundary values \(b_{\ast, N}\) converge to the same limits. Step 2 with these inputs gives the convergence of the sources and of \(\beta^N_q\), and Lemma 24.1 that of the multipliers \(\hat{\Theta}^{[j], N}_q (k)\).\(\Box\)

29Clipping and the complete drift

We keep all quadratic, covariance and scale-clipping terms. The exact gradient of the defect for \(q \geqslant q_{\ast}\), computed in Lemma 21.2, is

with \(D \Upsilon_q = D \left( - \frac{1}{2} \text{Tr} [H_q - H_q (0)] \right)\) the trace part. Write \(\Upsilon = \Upsilon_{q_{\ast}}\). Following Section 12.4, for \(0 < q < q_{\ast}\) the coefficients are frozen:

The term \(D \partial_q \tilde{h}_q\) and the mass derivative do not appear below \(q_{\ast}\), which is why \(b_q\) jumps at \(q_{\ast}\).

Proof. For \(q\) in a compact \(I\), the profile \(\kappa_q \phi\) and its finitely many smooth parameter perturbations stay a positive distance inside the unclipped field ball, since \(\sup | \tau | < 2\). On the lattice \(\kappa_q (\phi_N)\) and \(j_N (\kappa_q \phi)\) differ uniformly by \(O (N^{- 1})\), an error which the next derivative bound removes. The finite chain rule expresses clipped jets through the converging unclipped jets, multiplied by smooth functions of the external spatial variables; repeated slots give diagonal measures. Thus the clipped Hessians and third and fourth vertices converge weakly with their Schur bounds. Lemmas 26.2 and 26.3 identify the clipped inverses, giving (a) at each \(q\).

The scale derivative must also be retained. Use

The first term and its tested derivatives converge by the trace argument of Proposition 27.2; the second by the finite chain rule. The mass derivative \(M' = \beta_q I +\mathfrak{S}_q [M^{[2]}, h^{[2]}]^{\circ}\) is identified in Theorem 28.3. In the complete gradient (29.1), both covariance terms converge weakly after testing the open slot, by Lemma 26.4. All terms are bounded in \(L^{\infty}\) on this fixed bounded profile. Thus \(De^N_q (\phi_N)\) and its trace part converge weakly in \(L^2\) at each scale, which is (b).

The inverse turns this into a strong drift limit. If \(f_N\) is bounded and weakly convergent in \(H\), and \(R_N \to R\) in Hilbert–Schmidt norm, then \(R_N f_N - Rf = (R_N - R) f_N + R (f_N - f) \to 0\): the second term uses the compactness of \(R\). Apply this to \(- De^N\). For \(\varrho_q\), the quadratic part converges on smooth tests by Lemma 24.1; apply the same compact argument with \(C_q\) to the weakly converging clipped first derivative. This is (c).

The bounds \(\|H^N_q (y)\|_{\text{HS}}^2 \leqslant Ca_2 (q)\) and \(\|b^N_q (y)\|_2 \leqslant C \lambda q^{- 3 / 2} \|y\|_2\), of Lemma 12.1 and Proposition 12.2, and \(\| \varrho^N_q (y)\|_2 \leqslant \varepsilon_q \|y\|_2\) are uniform. So for a space–scale profile we integrate the convergence at each scale over \(I\) by dominated convergence.

For \(q < q_{\ast}\), (a) and (b) at the single scale \(q_{\ast}\) give \(H^N_{q_{\ast}} (\phi_N) \to H_{q_{\ast}} (\phi)\) and the weak convergence of \(D \Upsilon^N (\phi_N)\); adding the strongly convergent \(\phi_N\) and applying the same compact argument gives \(b^N_q (\phi_N) \to H_{q_{\ast}} (\phi) (\phi - D \Upsilon (\phi))\). The domination is (12.9).\(\Box\)

The proof gives the convergence at every fixed scale, not only in \(L^2 (I)\). This matters: for the frozen coefficients, the scale \(q_{\ast}\) is a single point of the coefficient interval.

Proof. Write \(\Delta H = H_q (y) - H_q (z)\), \(H_y = H_q (y)\), \(H_z = H_q (z)\), and \(\textsf{D}_y = D^2 \tilde{h}_q (y)\), \(\textsf{D}_z = D^2 \tilde{h}_q (z)\). The exact identity is

Every row of \(H_z\) has bounded \(L^2\) norm on \(I\), by the diagonal bound (12.3), so \(\|M_a H_z \|_{\text{HS}} \leqslant C_I \|a\|_2\). Also

Bound a row of the last inverse in this product by Minkowski, then collapse the third tensor over one internal slot: for a three-tensor \(T\), \(S (x, u) = \int |T (x, v, u) | \hspace{0.17em} \mathrm{d} v\) has both Schur masses at most \(s (T)\). Hence

where \(\textsf{S}_3 = \sup s (D^3 \tilde{h}_q)\) is finite by (12.2). With \(\|H_y \| \leqslant C / q\) this proves (29.3).

Let \(d (x) = \| \Delta H (x, \cdot)\|_2\), so \(\|d\|_2 = \| \Delta H\|_{\text{HS}}\). Since \(H^2 (u, v)\) is the inner product of the two rows, symmetry gives

In (29.1), split \(y \text{diag} H_y^2 - z \text{diag} H_z^2 = (y - z) \text{diag} H_y^2 + z \text{diag} (H_y^2 - H_z^2)\). For the third vertex term, the vertex difference is an integral of \(D^4 \tilde{h}_q [y - z]\), and the covariance difference is bounded using the last display and the rooted third tensor. The terms \(M'_q y\) and \(D \partial_q \tilde{h}_q\) are linear, respectively Lipschitz, with constant \(Cg_q\), by \(\|\mathfrak{S}_q \|_{\text{TV}} \leqslant Cg_q\) and (12.2). These give

Finally use

The global linear bound of Proposition 12.2 gives \(\|H_z De_q (z)\|_{\infty} = \|b_q (z)\|_{\infty} \leqslant C_I \|z\|_{\infty}\), and the second bound of (29.6) handles the remaining Hessian term. The coordinate estimate is the bound on \(D \varrho_q\) in (12.5).

For the frozen drift, \(b (y) - b (z) = H_y (y - z - D \Upsilon (y) + D \Upsilon (z)) - H_y [\lambda M_a + \textsf{D}_y - \textsf{D}_z] b (z)\), with \(H_y, \textsf{D}_y\) at scale \(q_{\ast}\) and \(\|b (z)\|_{\infty} \leqslant C \|z\|_{\infty}\) by (12.9). The same bounds apply.\(\Box\)

Proof. Fix \(q \geqslant q_{\ast}\) and \(L < \infty\). On the ball \(\|y\|_{\infty} \leqslant L\) we have \(\|y^2 - z^2 \|_2 \leqslant 2 L \|y - z\|_2\), so (29.3)–(29.5) give uniform Lipschitz continuity for the \(H\) norm, uniformly in \(N\). Smooth profiles bounded by \(L\) are dense in this ball, by mollification, and the ball is closed in \(H\). For smooth \(\phi, \phi'\) bounded by \(L\), the bounds pass to the limit from \(\phi_N, \phi'_N\), in (29.4) by the lower semicontinuity of the norm for weak limits. This defines the extension and gives the estimates.

For the convergence, a limit \(z\) of \(\iota_N z_N\) is bounded, as balls are closed. Take a smooth \(\phi\), bounded by \(L = \sup \|z_N \|_{\infty}\), with \(\| \phi - z\|_2 \leqslant \eta\). Since \(\| \iota_N z_N - \iota_N \phi_N \|_2 \leqslant \| \iota_N z_N - z\|_2 + \eta + \| \phi - \textsf{P}_N \phi \|_2\), the three term split

(with embeddings omitted) and Proposition 29.1 give the claim as \(N \to \infty\) and then \(\eta \to 0\). The same split, after testing with \(f \in H\), gives the weak convergence of \(De\) and \(D \Upsilon\), and without testing that of \(b\) and \(\varrho\).

The form of \(\varrho_q\) follows by the Lipschitz bound \(\|D \tilde{h}^N_q (y) - D \tilde{h}^N_q (z)\|_2 \leqslant s (D^2 \tilde{h}_q) \|y - z\|_2 \leqslant Cg_q q \|y - z\|_2\) and (12.2), passed to the weak limit, together with the fact that the multiplier \((q + 4 \pi^2 |k|^2)^{- 1} (\mu_q + \hat{\Theta}_q (k))\) is bounded.

For the measurability, on smooth \(\phi\) the maps \(q \mapsto H_q (\phi), b_q (\phi), \varrho_q (\phi)\) are limits of continuous functions of \(q\) (the second piecewise), so are Borel. For explicit jointly measurable versions, choose in each supremum ball a countable dense collection of smooth profiles and select the first profile within \(H\) distance \(2^{- n}\) of the input. This is a Borel selection. The continuity estimates identify the limit of the resulting countably valued maps. Supremum balls are closed in \(H\), so their increasing union gives Borel coefficient maps on bounded fields. Fixed values may be assigned outside this union, which does not affect the actual integrals. For \(q < q_{\ast}\) everything reduces to scale \(q_{\ast}\), and \(y \mapsto y\) is continuous.\(\Box\)

30Random inputs

The conclusion below is the statement of Section 18.1 of Lecture 7, used in the proofs of Theorems 18.4 and 18.6. We work on one probability space. All fields \(Y^N\) are measurable functions on \(\Omega \times I\) with values in \(H_N\), and \(Y\) is a random element of \(L^2 (I \times \mathbb{T}^3)\); \(I\) is a compact interval in \((0, \infty)\), and the coefficients for \(q < q_{\ast}\) are the frozen ones (29.2). If \(I\) contains \(q_{\ast}\) in its interior, split it there.

Proof. Bounded fields. Let \(L < \infty\). Consider nonrandom fields with \(\|z\|_{\infty} \leqslant L\) on \(I \times \mathbb{T}^3\). Integrating (29.3)–(29.5) over \(I\), on this set \(\|y^2 - z^2 \|_2 \leqslant 2 L \|y - z\|_2\) gives

and the same for the continuum maps of Proposition 29.3: uniform continuity in strong space–time \(L^2\). Smooth space–scale profiles bounded by \(L\) are dense in this set: convolve in space and scale, using a bounded extension at the scale endpoints. Cell averages preserve the supremum bound and converge strongly.

Reduction to a net. Fix \(\varepsilon > 0\). Choose \(L\) so that the probability that \(\sup_I \|Y^N \|_{\infty} > L\) is below \(\varepsilon\) for all \(N\). The closed ball \(S_L\) of radius \(L\) of \(L^{\infty} (I \times \mathbb{T}^3)\) is closed in \(L^2 (I \times \mathbb{T}^3)\), so the convergence in probability gives \(\mathbb{P} (Y \notin S_L) \leqslant \text{liminf} \mathbb{P} (\iota_N Y^N \notin S_L) \leqslant \varepsilon\). As \(\varepsilon\) is arbitrary, \(Y\) is almost surely bounded. The law of the single random element \(Y\) of the Polish space \(L^2 (I \times \mathbb{T}^3)\) is tight, so there is a compact \(\mathcal{S} \subset S_L\) with \(\mathbb{P} (Y \notin \mathcal{S}) \leqslant 2 \varepsilon\).

Given \(\eta > 0\), choose finitely many smooth space–scale profiles \(\phi^1, \ldots, \phi^m\), bounded by \(L\), so that every point of \(\mathcal{S}\) is within \(\eta\) of one of them in \(L^2 (I \times \mathbb{T}^3)\). On the event that \(Y \in \mathcal{S}\), that \(\sup_I \|Y^N \|_{\infty} \leqslant L\) and that \(\| \iota_N Y^N - Y\|_{L^2} \leqslant \eta\), whose probability is at least \(1 - 4 \varepsilon\) for large \(N\), pick \(i\) with \(\|Y - \phi^i \|_{L^2} \leqslant \eta\). Then

in \(L^2 (I ; \text{HS})\), with embeddings omitted. The first term is at most \(C_{I, L} (2 \eta + \text{max}_j \| \phi^j - \textsf{P}_N \phi^j \|_{L^2})\) by (30.2), the last at most \(C_{I, L} \eta\), and the middle term tends to zero by Proposition 29.1, deterministically. So the probability that the left side exceeds \(4 C_{I, L} \eta\) is at most \(5 \varepsilon\) for large \(N\). Letting \(\eta \to 0\) and then \(\varepsilon \to 0\) proves (30.1) for \(H\). The same argument applies to \(b\) and \(\varrho\). The limits are the maps of Proposition 29.3, evaluated at \(Y_q\) for almost every \(q\); these are measurable by the last assertion of that proposition.

The uniform hypothesis. Assume now that for almost every \(q \in I\), \(\| \iota_N Y^N_q - Y_q \|_{\infty} \to 0\) in probability, with \(\sup_I \|Y^N \|_{\infty}\) tight. By Fubini and dominated convergence, \(\mathbb{E} \int_I \text{min} (1, \| \iota_N Y^N_q - Y_q \|_{\infty}) \hspace{0.17em} \mathrm{d} q \to 0\). Along a subsequence, almost surely, \(\iota_N Y^N_q \to Y_q\) uniformly for almost every \(q\). Let \(A_N\) be the event that \(\sup_I \|Y^N \|_{\infty} \leqslant L\), with \(\mathbb{P} (A_N) \geqslant 1 - \varepsilon\). The event that \(A_N\) occurs infinitely often along the subsequence has probability at least \(1 - \varepsilon\), and on it \(\|Y_q \|_{\infty} \leqslant L\) for almost every \(q\). So \(\mathbb{P} (\|Y\|_{\infty} \leqslant L) \geqslant 1 - \varepsilon\), and on \(A_N\) intersected with this event,

whose expectation tends to zero. So \(\iota_N Y^N \to Y\) in \(L^2 (I \times \mathbb{T}^3)\) in probability, and the preceding part applies.

The frozen coefficients. The continuity and the consistency used above were established for the frozen drift in Lemma 29.2 and Proposition 29.1. Since \(\Upsilon^N (0) = 0\),

and this limit, for fields converging as in Proposition 29.3, is proved in Theorem 31.2 below. \(D \Upsilon\) is the trace part in the gradient (29.1) of \(e_{q_{\ast}}\), as in (12.8).\(\Box\)

Remark 30.2. (Which hypotheses are used) The theorem needs less than Section 18.1 of Lecture 7 assumes: uniform convergence in space is not used, only strong \(L^2 (I \times \mathbb{T}^3)\) convergence, and the sup bound need only be tight, with no moments. The estimate (29.5) requires no bound at all, and (29.3) requires only the \(L^4\) convergence of the field, but we do not use this. Moments of the sup norm such as those of Lemma 17.1 are one way of obtaining the tightness.

31The coordinate, high modes and stochastic integrals

We now prove Lemma 18.3 and the statement on stochastic integrals in Section 18.1 of Lecture 7. Let \(\tau_a\) denote the lattice translation by \(a \in \Lambda_N\).

Proof. Let \(\rho_{\delta}\) be a mollifier supported in the ball of radius \(\delta\). If \(z\) lies in the cell of \(x\) and \(|w| \leqslant \delta\), then \(z - w\) lies in the cell of a lattice point within \(\delta + C / N\) of \(x\). So \(\| \iota_N u_N - \rho_{\delta} \ast \iota_N u_N \|_{\infty} \leqslant \sup_{|a| \leqslant \delta + C / N} \| \tau_a u_N - u_N \|_{\infty}\), whose limsup is at most \(\varpi (2 \delta)\). For each fixed \(\delta\), the functions \(\rho_{\delta} \ast \iota_N u_N\) are bounded in \(C^1\), so by the Ascoli theorem and a diagonal argument over \(\delta = 1 / m\) they converge uniformly along a subsequence for every \(m\). By the last bound \(\iota_N u_N\) is uniformly Cauchy along it, and the limit is a uniform limit of the continuous limits of \(\rho_{1 / m} \ast \iota_N u_N\). If \(\iota_N u_N \to u\) in \(H\), then every such limit equals \(u\).\(\Box\)

The continuum coordinate. Applying \(C_s\), the coordinate equation \(K_s y + D \bar{\Gamma}_s (y) = K_s x\) reads, by (12.5),

We take this equation, with the maps of Proposition 29.3, as the continuum coordinate equation, for \(y \in L^{\infty}\). For \(s < q_{\ast}\) we use (12.7), that is \(y_s (x) = y_{q_{\ast}} (x - (q_{\ast} - s) C_{q_{\ast}} x)\).

The continuum defect is defined on \(L^{\infty}\) by

Theorem 31.2 shows that it is the limit of the lattice defects.

Proof. Let first \(s \geqslant q_{\ast}\).

Uniqueness. Let \(y, y' \in L^{\infty}\) solve (31.1), and put \(\delta = y - y'\), \(f = D \tilde{h}_s (y) - D \tilde{h}_s (y')\) and \(w = \frac{\lambda}{3} (y^3 - (y')^3)\), which lie in \(H\). By the form of \(\varrho_s\) in Proposition 29.3, in Fourier variables,

By the small relative form bound used in the proof of Lemma 26.1, \(m_s (k) \geqslant \frac{1}{2} (s + 4 \pi^2 |k|^2)\). So \(\delta \in H^2\), and \(y - x \in H^1\) by (31.1). Multiplying by the conjugate of \(\hat{\delta} (k)\) and summing,

as \(\langle \delta, w \rangle \geqslant 0\) by the monotonicity of the cube. For \(q_{\ast}\) large this gives \(\delta = 0\).

Convergence. Put \(y_N = y^N_s (x_N)\). As \(\bar{\Gamma}^N_s\) is even, \(y^N_s (0) = 0\), so Lemma 16.3 gives \(\|y_N \|_{\infty} \leqslant 3 \|x_N \|_{\infty}\), which is bounded. As \(\bar{\Gamma}^N_s\) is translation invariant, the uniqueness of the coordinate gives \(y^N_s (\tau_a x_N) = \tau_a y^N_s (x_N)\), so the same lemma gives

with \(\varpi_x\) the modulus of continuity of \(x\). By Lemma 31.1, along a subsequence \(\iota_N y_N \to y\) uniformly, with \(y\) continuous. Pass to the limit in the lattice form of (31.1). Then \(\iota_N (y_N^3) \to y^3\) uniformly and \(\iota_N C^N_s j_N \to C_s\) in \(\text{HS}\), by Lemma 26.1 with vanishing path, so that \(\iota_N C^N_s y_N^3 \to C_s y^3\) in \(H\). By (29.7), \(\iota_N \varrho^N_s (y_N) \to \varrho_s (y)\) in \(H\). Thus \(y\) solves (31.1). So a bounded solution exists, it is continuous, and by uniqueness every subsequential limit equals it: the whole sequence converges. The Lipschitz bound passes to the limit, using \(x_N = j_N x\) and \(x'_N = j_N x'\).

Defect. The lattice defect satisfies \(e^N_s (0) = 0\), as \(\tilde{h}^N_s\) and its scale derivative vanish at \(0\), and so \(e^N_s (y) = \int_0^1 \langle De^N_s (ty), y \rangle \hspace{0.17em} \mathrm{d} t\). Let \(z_N, z\) be as in (b). For each \(t\), \(\iota_N (tz_N) \to tz\) in \(H\) and is bounded, so by Proposition 29.3, \(\iota_N De^N_s (tz_N) \to De_s (tz)\) weakly in \(H\), and \(\iota_N z_N \to z\) strongly; the pairing converges. As \(De^N_s (0) = 0\) by evenness, (29.4) with second field \(0\) gives \(\|De^N_s (tz_N)\|_2 \leqslant C (\|z_N \|_2 +\|z_N^2 \|_2)\), so dominated convergence in \(t\) gives \(e^N_s (z_N) \to e_s (z)\). The same proof applies to \(\Upsilon^N (z_N)\) with \(D \Upsilon^N\), by (30.3). This also proves the limit claim in (30.3). Part (a) makes the choice \(z_N = y^N_s (x_N)\) admissible.

The bound (c). The proof of the global linear bound of Proposition 12.2 (Lemma 21.2) gives \(\|De^N_s (y)\|_p \leqslant Cg_s \|y\|_p\) for all \(y\), and so \(|e^N_s (y) | \leqslant Cg_s \|y\|_2^2\). For \(s < q_{\ast}\) this is (12.9).

The frozen case \(s < q_{\ast}\). By (12.7), \(y^N_s (x_N) = y^N_{q_{\ast}} (x_N')\) with \(x_N' = x_N - (q_{\ast} - s) C^N_{q_{\ast}} x_N\). The operator \(C^N_{q_{\ast}}\) has norm \(1 / q_{\ast}\) on \(L^{\infty}\) and commutes with lattice translations, so \(u_N = C^N_{q_{\ast}} x_N\) is bounded and \(\| \tau_a u_N - u_N \|_{\infty} \leqslant q_{\ast}^{- 1} \| \tau_a x_N - x_N \|_{\infty}\). Also \(\iota_N u_N \to C_{q_{\ast}} x\) in \(H\) by Lemma 26.1. So by Lemma 31.1, \(\iota_N x_N' \to x' = x - (q_{\ast} - s) C_{q_{\ast}} x\) uniformly, with \(x'\) continuous. The case \(s = q_{\ast}\) gives (a), with Lipschitz constant \(3 \|I - (q_{\ast} - s) C_{q_{\ast}} \|_{\infty \to \infty} \leqslant 6\). By definition \(e^N_s (y) = - \frac{1}{2} \|y\|_2^2 + \Upsilon^N (y)\); \(\| \iota_N z_N \|_2^2 \to \|z\|_2^2\) by the strong convergence, and \(\Upsilon^N (z_N) \to \Upsilon (z)\) as shown. This proves (b).\(\Box\)

The following gives the second claim of Section 18.1 of Lecture 7: the identity \(H^N = (I - H^N D^2 \bar{\Gamma}^N) C^N\) and its consequence on high modes. Let \(F^N_{> m}\) be the Fourier projection on \(H_N\) onto the modes \(k \in \Lambda^{\ast}_N\) with \(|k| > m\), and \(F_{\leqslant m}\) the Fourier projection on \(H\) onto the modes \(|k| \leqslant m\).

Proof. The identity is \(H^N (K_q + D^2 \bar{\Gamma}^N) = I\). By (12.1), \(D^2 \bar{\Gamma}^N = \lambda y^2 + \mu_q + \Theta_q + D^2 \tilde{h}_q (y)\). By Lemma 12.1, \(\|H^N \|_{2 \to 2} \leqslant C / q\). So \(H^N (\lambda y^2 + \mu_q + D^2 \tilde{h}_q)\) has norm at most \(C (\lambda L^2 / q + g_q)\), by (11.12) and (12.2). The kernel \(\Theta_q\) is not bounded on \(L^2\) uniformly in \(N\), and we argue by the other resolvent identity. By Lemma 24.1, \(\Theta_q C_q\) has the multiplier \(\hat{\Theta}_q (k) (q + c_k)^{- 1}\), of size at most \(Cv_1 (\Theta_q) q^{- 1 / 2} \leqslant CR \lambda^2 q^{- 1} \leqslant \frac{1}{2}\). From \(H^N = C_q - C_q D^2 \bar{\Gamma}^N H^N\),

so that \(\| \Theta_q H^N \|_{2 \to 2} \leqslant C \lambda^2 q^{- 1} (1 + \lambda L^2 / q)\), and \(H^N \Theta_q\) is its adjoint. This proves (31.3). Next

and multiplying by the bounded factor of (31.3) gives (31.4). The frozen case uses the same identity at \(q_{\ast}\).\(\Box\)

The projected noise. Let \(W\) be a cylindrical Brownian motion on \(H\). As in Section 18.3 of Lecture 7, its Fourier projection is \(W^N = E_N W\), where

a real operator \(H \to H_N\) with \(E_N E_N^{\ast} = I\), as \(N\) is odd and the \(e^N_k\) are orthonormal in \(H_N\). The alternative is \(W^N = j_N W\), with \(j_N j_N^{\ast} = j_N \iota_N = I\). Both are cylindrical on \(H_N\). We show that both give the same limit.

Proof. Let \(E^{\sharp}_N\) be either \(E_N\) or \(j_N\). Then \(\int \iota_N H^N (Y^N) \hspace{0.17em} \mathrm{d} W^N = \int \mathcal{J}^N \hspace{0.17em} \mathrm{d} W\), with the integrand \(\mathcal{J}^N_q = \iota_N H^N_q (Y^N_q) E^{\sharp}_N\), which is progressively measurable. So by Doob's inequality and the Itô isometry the left side of (31.5) is at most \(4\mathbb{E} \int_I \| \Phi^N_q - H_q (Y_q)\|_{\text{HS}}^2 \hspace{0.17em} \mathrm{d} q\). By (12.3) and (12.9), \(\| \Phi^N_q \|_{\text{HS}}^2 \leqslant \text{Tr} H^N_q (Y^N_q)^2 \leqslant C_I\) deterministically, and the same bound holds for the limit by lower semicontinuity. It is therefore enough to show that \(\Phi^N \to H (Y)\) in \(L^2 (I ; \text{HS})\) in probability.

For \(E^{\sharp}_N = j_N\) this is Theorem 30.1. For \(E^{\sharp}_N = E_N\), it remains to show that \(\| \iota_N H^N_q (Y^N_q) (E_N - j_N)\|_{\text{HS}} \to 0\) in \(L^2 (I)\), in probability. Work on the event that \(\sup_I \|Y^N \|_{\infty} \leqslant L\) and split with \(F_{\leqslant m}\) and \(I - F_{\leqslant m}\).

Low modes. \(E_N e_k = e^N_k\), while \(j_N e_k =\mathfrak{s}_N (k) e^N_k\) with \(\mathfrak{s}_N (k) = \prod_i \text{sinc} (\pi k_i / N)\). So for \(m\) fixed, \(\|(E_N - j_N) F_{\leqslant m} \|_{\text{HS}} \to 0\), and \(\|H^N \|_{2 \to 2} \leqslant C\).

High modes, Fourier. Since \(E_N (I - F_{\leqslant m}) = F^N_{> m} E_N\) and \(\|E_N \| \leqslant 1\), (31.4) gives \(\|H^N E_N (I - F_{\leqslant m})\|_{\text{HS}} \leqslant C_L m^{- 1 / 2}\), uniformly in \(N\) and in \(q \in I\).

High modes, cell average. This operator does not preserve high modes exactly, because of aliases, but

The first term tends to \(0\) in \(L^2 (I)\) in probability by Theorem 30.1. The second tends to \(0\) as \(m \to \infty\), for each \(q\), and it is dominated, so it tends to \(0\) in \(L^2 (I)\) almost surely. Sending \(N \to \infty\) and then \(m \to \infty\) and \(L \to \infty\) gives the claim.

For the final assertions, the frozen case is included in Theorem 30.1. If \(I\) contains \(q_{\ast}\), split it there; the integrand is defined for all \(q\), and the formula for \(H_q\) is continuous across \(q_{\ast}\), as the drift is not involved.\(\Box\)

Remark 31.5. (Across the auxiliary scale) \(\bar{\Gamma}_q\) is continuous in \(q\) at \(q_{\ast}\), and \(y_{q_{\ast} -} = y_{q_{\ast}}\) by (12.7), as \(C_{q_{\ast}} K_{q_{\ast}} = I\). So \(Y_q = y_q (x_q)\) is continuous across \(q_{\ast}\), though \(b_q\) jumps there. Its forward equation is obtained by integrating on each side and adding; the jump of \(b\) does not matter for convergence in \(L^2 (I ; H)\). The adaptedness of \(Y^N\) and \(Y\) in Theorem 31.4 is a genuine hypothesis, not implied by the convergence. In Section 18.3 of Lecture 7 it holds since the explicit flow is a strong solution, adapted to \(W\).

32Complements not used in the lectures

The lectures obtain the limit law from an explicit density, and the FBSDE from it by Girsanov's theorem. This section records the older routes and some stronger statements. References to the lectures are to their present numbering. In the subsection on compactness, the convergence is the stronger one of \(C (I ; H)\) and strong \(L^2 (I ; \text{HS})\), which the lectures do not claim. These routes predate the density of Section 18 of Lecture 7; where they differ from it, the lectures are authoritative.

In this section \(V_q\) without argument is the field (13.4), and \(V_q (x)\) the potential; \(\chi_q (y)\) is the pointwise large-field indicator, while \(\chi_q (x)\) in the uniqueness proofs is a cut-off of the norm and \(\chi\) in the penalty a profile; \(\tau\) in the cut-offs is not the clip profile; \(\xi\) denotes the free flow or \(Y - y\) as stated; \(A\) is the penalty constant, \(A (q, x)\) the Feynman–Kac exponent; \(J\) is \(D \kappa\), \(J_q\) the filter, \(J_{\infty}\) the energy.

32.1Regularity in space and scale

On a finite scale interval \(I = [q_{\ast}, Q]\) write the field equation (13.5) as the linear elliptic equation

The energies control \(\mathbb{E} \int_I \|Y_q \|_6^6 \mathrm{d} q\) (since \(|Y|^6 \leqslant r_q^6 + |t_q |^6\)) and \(\mathbb{E} \int_I \|Z_q \|_2^2 \mathrm{d} q\), and \(\mathbb{E} \| \zeta_q \|_2^2 \leqslant C \sqrt{q}\). Hence \(\mathbb{E} \int_I \|f_q \|_2^2 \leqslant C_I\), and testing with \(V\),

So the part \(V\) of the approximate forecast is in \(H^1\), and \(Y = V + \zeta - \varrho (Y)\) inherits the regularity of the noise \(\zeta\). This is that of the free field, whatever the interaction: by Lemma 13.2, \(\mathbb{E} \| \zeta_q \|_{H^{\sigma}}^2 \leqslant C (q - q_0) \sum_k (1 + |k|^2)^{\sigma} (q + c|k|^2)^{- 2}\), finite uniformly in \(N\) for \(\sigma < \frac{1}{2}\).

The spatial bound combines the \(H^1\) bound on \(V\), the \(H^{\sigma}\) regularity of \(\zeta\) (\(\sigma < \frac{1}{2}\), for which the cell embedding is uniformly bounded), and the contraction property of \(\varrho\): since \(\varrho\) commutes with lattice translations \(\tau_a\), (12.5) gives \(\| \tau_a \varrho (Y) - \varrho (Y)\|_2 \leqslant \varepsilon_q \| \tau_a Y - Y\|_2\), so the increments of \(Y\) are controlled by those of \(V + \zeta\); the scale regularity uses the forward equation, whose drift \(b (Y) + H (Y) Z\) has bounded integrated square by Proposition 12.2 and the energies, and whose martingale part has bounded bracket density. The supremum bound of Lemma 17.1 upgrades \(L^2\) compactness to every \(L^p\).

32.2The limit by compactness: the noise

The delicate point is the backward martingale. Fields converging in law do not imply that the stochastic integrals converge to stochastic integrals of the limits.

Remark 32.2. (Hidden noise) Let \(B\) be a real Brownian motion and \(L^n_t = \int_0^t (1 + \sin (ns)) \mathrm{d} B_s\). Then \((L^n, B)\) converges in law to \((B + 2^{- 1 / 2} \tilde{B}, B)\), with \(\tilde{B}\) an independent Brownian motion: the integrands converge weakly to \(1\), but their squares converge weakly to \(\frac{3}{2}\). A limiting FBSDE obtained this way would have an extra martingale, orthogonal to the driving noise.

To exclude this one needs strong compactness of the integrands. For the forward equation the integrand \(H (Y)\) is an explicit function of \(Y\), and strong convergence follows from the coefficient limits. For the backward equation the integrand is \(\Xi = \text{Cov} - H (Y)\), and we need strong compactness of the posterior covariance \(\text{Cov}_q\). It comes from its exact dynamics.

Proof. The conditional law is an exponential family in \(p = K_q \Phi_q\) (proof of Proposition 10.4 of Lecture 6), with log-partition function \(\mathcal{K}_q (p) = \log \int \mathrm{e}^{\langle p, \varphi \rangle - \frac{q - q_0}{2} \| \varphi \|_2^2} \nu_N (\mathrm{d} \varphi)\), so \(\text{Cov}_q = D^2 \mathcal{K}_q (p)\) and \(\mathfrak{k}_q = D^3 \mathcal{K}_q (p)\). Differentiating, \(\partial_q \mathcal{K}+ \frac{1}{2} \text{Tr} D^2 \mathcal{K}+ \frac{1}{2} \|D\mathcal{K}\|_2^2 = 0\); differentiate this twice in \(p\): \((\partial_q + \frac{1}{2} \Delta_p + D\mathcal{K} \cdot D_p) D^2 \mathcal{K}= - (D^2 \mathcal{K})^2\), and apply Itô's formula along \(\mathrm{d} p = D\mathcal{K} (p) \mathrm{d} q + \mathrm{d} W\), which follows from (10.6), \(\partial_q K_q = I\) and \(\psi_q = \Phi_q + C_q F_q\).\(\Box\)

This is (8.9) of Lecture 4 for \(\partial_q \mathcal{R}= 1\): conditional variance is resolved at the rate \(\text{Cov}^2\). For a smooth test \(f\), the process \(\langle f, \text{Cov}_q f \rangle \geqslant 0\) is a nonnegative supermartingale, with drift \(- \| \text{Cov}_q f\|^2\), and initial value \(\text{Var}_{\nu_N} \langle \Phi_{\infty}, f \rangle\), bounded uniformly in \(N\) by Corollary 17.2. A nonnegative supermartingale has controlled oscillations: after stopping at the first time it exceeds a level \(K\) (an event of probability at most \(C_f / K\)), its decreasing part \(A\) and the bracket of its martingale part \(J\) satisfy \(\mathbb{E}A_Q \leqslant C_f\) and \(\mathbb{E} [J]_Q \leqslant K^2 + 2 KC_f\), which gives time regularity in \(W^{a, 1}\), \(a < \frac{1}{2}\), and hence strong compactness of each finite block \(\langle f_i, \text{Cov} f_j \rangle\). In this subsection \(I = [q_0, Q]\), since \(\psi\), \(W\) and the covariance do not involve \(Y\). Spatial tails come from the isometry (recall \(\psi_{q_0} =\mathbb{E} \Phi_{\infty} = 0\), \(\nu_N\) being even), with symmetry exchanging input and output tails, \(\mathbb{E} \int_I \|(I - \textsf{F}_m) \text{Cov}_q \|_{\text{HS}}^2 \mathrm{d} q =\mathbb{E} \|(I - \textsf{F}_m) \psi_Q \|_2^2\), and from the terminal averaging \(\psi_Q =\mathbb{E}_Q \int_L^{L + 1} (Y_s + Z_s) \mathrm{d} s\), using the \(H^{\sigma}\) bound of Proposition 32.1 and \(\mathbb{E} \|Z_s \|_2^2 \leqslant Cs^{- 5 / 2}\) (first \(m \to \infty\), then \(L \to \infty\)). Altogether \(\text{Cov}^N\) is tight in strong \(L^2 (I ; \text{HS})\). Since \(S \mapsto \int SS^{\ast}\) is continuous from \(L^2 (I ; \text{HS})\) to \(C (I ; \text{trace class})\), the brackets of \(\psi^N\) are tight in \(C (I)\); a continuous martingale with tight brackets and uniformly small maximal Fourier tails (from the terminal averaging above and Doob's inequality) is tight in \(C (I ; H)\). Hence \(\psi^N\), and \(Z^N = \psi^N - Y^N\), are tight in \(C (I ; H)\). Strong convergence of the integrands gives convergence of the brackets \(\int \text{CovCov}^{\ast}\) and of the covariations with \(W\), and then:

The argument is: the limiting \(\psi\) is a continuous square-integrable martingale for the joint filtration (fixed-time martingale identities pass to the limit with uniform integrability); its bracket and its covariation with \(W\) are those of \(\int \text{Cov} \hspace{0.17em} \mathrm{d} W\); so \(\psi - \int \text{Cov} \hspace{0.17em} \mathrm{d} W\) has zero bracket and vanishes. No extra martingale survives. The same argument for the forward equation identifies its noise with the same \(W\), and identifies the limit of the auxiliary Gaussian field \(X\) as \(\int C \hspace{0.17em} \mathrm{d} W\), not a new Gaussian field.

32.3The limit by compactness: the result

Below the auxiliary scale. Above \(q_{\ast}\) the proof is that of Section 32.2, Propositions 32.1 and 32.4, with the moments of \(Y_{q_{\ast}}\) from Section 17, and spatial compactness at \(q_{\ast}\) by one-sided averaging on \([q_{\ast}, q_{\ast} + \delta]\). The processes \(\psi, W, \text{Cov}\) never involve \(Y\): for \(q \leqslant Q\), \(\psi_q =\mathbb{E}_q \psi_Q\) gives their moments, and the Riccati argument starts at \(q_0\) from the deterministic \(\text{Var}_{\nu_N} \langle \Phi^N_{\infty}, j_N f \rangle\). On \(I = [q_0, q_{\ast}]\), \(C_{q_{\ast}} K_s \Phi^N_s = C_{q_{\ast}} W^N_s + \int_{q_0}^s C_{q_{\ast}} \psi^N_u \hspace{0.17em} \mathrm{d} u\) is tight in \(C (I ; L^{\infty})\): the first term by Kolmogorov, the second as \(C_{q_{\ast}} : L^2 \to C^{1 / 2}\) and \(\psi^N\) converges in \(C (I ; H)\). By (12.7) and Lemmas 16.3, 18.3, \(Y^N_s = y^N_{q_{\ast}} (C_{q_{\ast}} K_s \Phi^N_s) \to Y\) uniformly on \(I \times \mathbb{T}^3\), so \(Z^N \to Z\) in \(C (I ; H)\). The convergence stated at the end of Section 18.1, at the single scale \(q_{\ast}\), gives \(H (Y^N) \to H (Y)\) in \(L^2 (I ; \text{HS})\) and \(b (Y^N) \to b (Y)\) in \(L^2 (I ; H)\), so \(\Xi^N \to \Xi\) and the limit solves the system on \(I\). At \(q_0\), \(\Phi_{q_0} = 0\): so \(Y_{q_0} = 0\) since \(\bar{\mathcal{F}}\) is odd, and \(Z_{q_0} = \psi_{q_0} =\mathbb{E}_{\nu} \Phi = 0\).

So the posterior structure survives the continuum limit: \(\psi = Y + Z\) is a closed observable, the conditional mean of the terminal field; \(Y\) is computed from the explicit action; and the error \(Z\) is small at high resolution. This is the FBSDE of Lecture 4, in forecast coordinates, for the limit \(\nu\) of the renormalised lattice measures, with the full noise retained.

32.4Density at a fixed scale

The following is not used in the lectures: it gives the law at a fixed high scale, and it is where the second chaos of \(\bar{V}_q\) becomes visible.

Proof. Bounds on \(\bar{V}\). On the unit volume, \(\bar{V}_q (x) = \min_y \left[ \bar{\Gamma}_q (y) + \frac{1}{2} \langle x - y, K_q (x - y) \rangle \right]\), the functional being strictly convex for \(q \geqslant q_{\ast}\). As \(U + \frac{1}{2} \mu_q y^2 + \tilde{h}_q \geqslant - C_q\) (the quartic dominates, as \(\textsf{S}_4 (h_q)\) is small, and \(| \tilde{h}_q (y) | \leqslant C \textsf{S}_4 (h_q) \|y\|_4^4\)), \(\bar{V}_q (x) \geqslant - C_q + \frac{1}{2} \langle x, \Theta_q' x \rangle\), with \(\Theta_q' = K_q (K_q + \Theta_q)^{- 1} \Theta_q\). With \(c_N = \frac{1}{2} \text{Tr} (Q_q \Theta'_q)\), the determinant formula for the second chaos gives the \(L^p\) bound, since the operator has norm \(\leqslant \frac{1}{2}\) and is Hilbert–Schmidt, uniformly in \(N\); here the multiplier of \(\Theta_q\) is at most \(C |k|\) and that of \(Q_q\) at most \(C |k|^{- 4}\). Convergence in probability is Step 4 of the proof of Theorem 32.13, without \(\chi_q\): the second chaos in \(L^2\), the other terms by continuity for the uniform norm.

Cut-off. For fixed \(N\), the law of \(\Phi_q\) is proportional to \(\mathrm{e}^{- \bar{V}_q + c_N - A_N (q, \cdot)} \gamma_{Q_q}\) by the definition of \(A_N\). For \(\chi_R\) a continuous cut-off of \(\|x\|_{\infty}\), on its support \(A_N\) is bounded and continuously convergent (Theorem 18.4), and \(\iota_N x_N \to x\) uniformly in probability; with the \(L^p\) bound,

with \(I (R) > 0\), and the same with a bounded continuous function under the integral. By Step 1 of the proof of Theorem 32.13, that is (12.5) and Lemma 17.1, \(\sup_N \mathbb{P} (\| \Phi^N_q \|_{\infty} > R / 2) \to 0\) as \(R \to \infty\). So for \(R\) large, \(I_N (R) / Z_N \geqslant \frac{1}{2}\), where \(Z_N\) is the total mass, and hence \(\text{limsup}_N Z_N \leqslant 2 I (R) < \infty\); \(\rho_q\) is finite by Fatou. Letting \(R \to \infty\) the law of \(\Phi_q^N\) converges to \(\rho_q\) normalised, as does \(Z_N\) to its total mass.

The formula. The kernel form follows by disintegrating \(\nu_N\) at scale \(q\), and using (18.12) with the uniform convergence in probability: the cylinder function \(\Pi^N_q O (x_N)\) is bounded, and converges for continuously convergent arguments. So \(\int O \hspace{0.17em} \mathrm{d} \nu = \int \Pi_q O \hspace{0.17em} \mathrm{d} \rho_q / \rho_q (1)\), which by (18.12) and (32.5) is (32.6), the factor \(\mathrm{e}^A\) cancelling.\(\Box\)

32.5Feynman–Kac comparison along the explicit flow

Let \(\bar{V}_q (x) = \Gamma^{\sharp} (y) + \frac{1}{2} \langle x - y, K_q (x - y) \rangle\), where \(y = y_q (x)\) solves \(K_q y + D \Gamma^{\sharp} (y) = K_q x\), for \(\Gamma^{\sharp}\) one of the actions below. Let \(e^{\sharp} = \partial_q \Gamma^{\sharp} - \frac{1}{2} \text{Tr} [H^{\sharp} - H^{\sharp} (0)]\) and \(c (q) = \frac{1}{2} \text{Tr} (C_q - H_q (0))\).

Proof. By the envelope theorem \(\partial_q \bar{V} = \partial_q \Gamma^{\sharp} (y) + \frac{1}{2} \|x - y\|_2^2\). Further \(D \bar{V} = K_q (x - y)\), and since \(Dy = H^{\sharp} K_q\), \(D^2 \bar{V} = K_q - K_q H^{\sharp} K_q\). So \(\text{Tr} (C_q^2 D^2 \bar{V}) = \text{Tr} (C_q - H^{\sharp})\) and \(\|C_q D \bar{V} \|_2 = \|x - y\|_2\).\(\Box\)

The penalty. Let \(\chi \in C^{1, 1}\), smooth off \(\frac{15}{16}\) and nondecreasing, with \(\chi (u) = 0\) for \(u \leqslant \frac{15}{16}\), \(\chi (u) = 1\) for \(u \geqslant 1\), and \(\chi (u)^2 = \left( u - \frac{15}{16} \right)^4\) near \(\frac{15}{16}\). For a constant \(A\) put \(\mathfrak{g}_s (y) = g_s \int y^2 \chi^2 (|y| / r_s)\) and

\(F_q\) is even and \(C^4\), vanishes on \(|u| \leqslant \frac{15}{16} r_q\), and the \(s\) integral runs up to \(s_u = \frac{256}{225} \lambda u^2\). Let \(v = |u| / r_q - \frac{15}{16}\). Then for \(v > 0\), with constants independent of \(A\) and \(q\),

the second from differentiating \(u^2 \int_q^{s_u} \lambda s^{- 1 / 2} \chi^2 (|u| / r_s) \hspace{0.17em} \mathrm{d} s\) twice in \(u\): near the threshold the integral is of order \(u^2 g_q qv^5\), and for \(|u| \gg r_q\) it is of order \(A \lambda^{3 / 2} |u|^3\).

Proof. \(F'' \leqslant CAg_q \lambda y^2\) on its support, as \(q \leqslant C \lambda y^2\) there. So \(\lambda y^2 \pm F'' (y) \geqslant 0\) for \(Ag_q\) small, and the proof of Lemma 12.1 applies with this potential: killing gives \(0 \leqslant (K_q + \lambda y^2 \pm F'')^{- 1} \leqslant C_q\) entrywise, and the Neumann series handles \(M + D^2 \tilde{h}\). For the drift, the new terms in \(De^{\pm}\) are \(\mp AD\mathfrak{g}_q\) with \(|D\mathfrak{g}_q | \leqslant Cg_q |y|\), and \(\pm \frac{1}{2} F''' (y) \text{diag} (H^{\pm})^2\) together with the change of \(H\) in the terms of \(De\). Since \(|F''' (u) | \leqslant CA \lambda^{3 / 2} 1_{v > 0}\) and \(\text{diag} (H^{\pm})^2 \leqslant Ca_2\), the former is at most \(CAg_q^2 |y|\). Multiplication by \(H^{\pm}\) costs \(C / q\), and the global linear bound for \(b\) gives the rest.\(\Box\)

The linear large-field bound. Write \(De = D \kappa De^{\text{o}} \circ \kappa +\mathfrak{D}\) as in Lemma 21.2, \(\mathfrak{D}= De - D \kappa De^{\text{o}} \circ \kappa\).

Proof. The ledger (21.10) has six rows, of which four are linear in \(t\). Only rows four and six carry \(y^2 \chi\), through \(\mathcal{D}_2 = H^2 {- H^{\text{o}}}^2 = - H^2 \textsf{D} H^{\text{o}} - H \textsf{D} {H^{\text{o}}}^2\) with \(\textsf{D} = \lambda M_{y^2 - z^2} + \textsf{D}'\), where \(z\) is now the clipped field and \(\textsf{D}'\) the nonlocal insertions. The latter cost \(Cgq \| \chi \|_2\), and \(r \chi \leqslant |t|\). For the local insertion, put \(e_0 = (y^2 - z^2) / |y| \leqslant C |t|\) and \(Q = \lambda^{1 / 2} M_{|y|} H\). By Lemma 21.1, \(\|Q\|_{2 \to 2} \leqslant Cq^{- 1 / 2}\) and \(\text{diag} H^2, \text{diag} {H^{\text{o}}}^2 \leqslant Ca_2\). Let \(T\) be a symmetric \(3\)-tensor and \(\varphi \in L^2\). Then \(X = H^2 \lambda M_{y^2 - z^2} H^{\text{o}}\) reads \(\lambda^{1 / 2} (HQ^{\ast}) M_{e_0} H^{\text{o}}\), and Cauchy–Schwarz in all variables gives

Integrating \(a\) and using \(HQ^{\ast} QH \leqslant \|Q\|^2 H^2\), the first factor is at most \((Ca_2 s (T) / q)^{1 / 2} \| \varphi \|_2\); integrating \(x, u\) and then \(v\), the second is at most \((Ca_2 s (T))^{1 / 2} \|e_0 \|_2\). So \(\| \text{Ctr}_X T\|_2 \leqslant Cr^{- 1} a_2 s (T) \|t\|_2\). For \(X = H \lambda M_{y^2 - z^2} {H^{\text{o}}}^2\) write it as \(\lambda^{1 / 2} (Q^{\ast} M_{e_0} H^{\text{o}}) H^{\text{o}}\) and put \(\varphi^2\) on the factor with the outer \(H^{\text{o}}\): the same bound results. With \(s (f_3) \leqslant Cgq / r\) this is row six, \(Cg^2 \|t\|_2\). With the local tensor, \(s (T) = 1\), it gives \(\| \text{diag} \mathcal{D}_2 \|_2 \leqslant Cr^{- 1} a_2 \|t\|_2\), hence row four, \(C \lambda ra_2 r^{- 1} \|t\|_2 \leqslant Cg \|t\|_2\).\(\Box\)

Proof. The action \(\bar{\Gamma}\). On \(\|z\|_{\infty} \leqslant r_s\) the clip is the identity, so \(e_s (\pi_s y) = e_s^{\text{o}} (\pi_s y)\), and by evenness, the zero scalar jets and (11.14), \(|e_s^{\text{o}} (\pi_s y) | \leqslant C \lambda^5 s^{- 3} \| \pi_s y\|_4^4 + C \lambda^4 s^{- 2} \| \pi_s y\|_2^2\), the quadratic term coming from the centred quadratic jet \(\frac{1}{2} \langle y, \mathcal{E}^c_s y \rangle\). With \(z_{\theta} = \pi y + \theta (y - \pi y)\),

The vector \(y - \pi y\) is supported on the set \(\mathcal{A}\) where \(|y| > r_s\) and its norm is at most \(\|t_s (y)\|_2\); also \(\|t_s (z_{\theta})\|_2 \leqslant \|t_s (y)\|_2\). The first term of \(De\) is bounded pointwise by \(\sup |De^{\text{o}} | \leqslant \textsf{S}_4 (e^{\text{o}}) (2 r_s)^3 / 6 + \|\mathcal{E}^c_s \|_{\text{row}} 2 r_s \leqslant C \lambda^{7 / 2} s^{- 3 / 2}\), and \(|\mathcal{A}|^{1 / 2} \leqslant r_s^{- 1} \|t_s \|_2\), so it contributes at most \(C \lambda^4 s^{- 2} \|t_s \|_2^2 \leqslant Cg_s \|t_s \|_2^2\). By Lemma 32.9 so does the second. Since \(\chi = 1\) on \(\mathcal{A}\), \(g_s \|t_s \|_2^2 \leqslant \mathfrak{g}_s (y)\). Thus \(|e_s (y) | \leqslant C \lambda^5 s^{- 3} \| \pi_s y\|_4^4 + C \lambda^4 s^{- 2} \| \pi_s y\|_2^2 + C_1 \mathfrak{g}_s (y)\) with \(C_1\) independent of \(A\).

The penalty. \(\partial_s \delta_s = - A\mathfrak{g}_s\), so \(e^{\pm} = e \mp A\mathfrak{g}_s - \frac{1}{2} \text{Tr} (H^{\pm} - H)\). As \(H^{\pm} - H = \mp H^{\pm} D^2 \delta H\) and \(D^2 \delta\) is diagonal, \(| \text{Tr} (H^{\pm} - H) | \leqslant Ca_2 \int |F_s'' (y) |\). Pointwise, with \(u = y (x)\) and \(v\) as above: by (32.7), the ratio of \(a_2 |F'' (u) |\) to the density \(Ag_s u^2 \chi^2 (|u| / r_s)\) of \(A\mathfrak{g}_s\) is at most \(Cg_s / \min (v, 1)\). So for \(v \geqslant C_2 g_s\), this term is at most half of the density of \(A\mathfrak{g}_s\). For \(0 < v < C_2 g_s\), the thin layer, \(a_2 |F'' | \leqslant CAq^{- 1 / 2} q^{1 / 2} \lambda v^3 \leqslant CA \lambda g_s^3\). There \(| \pi_s u| \geqslant \frac{15}{16} r_s\), and \(\lambda^5 s^{- 3} r_s^4 = \lambda^3 s^{- 1}\), while \(A \lambda g_s^3 = A \lambda^4 s^{- 3 / 2} \leqslant \lambda^3 s^{- 1}\) for \(s\) large depending on \(A\). Hence \(\frac{1}{2} | \text{Tr} (H^{\pm} - H) | \leqslant \frac{A}{2} \mathfrak{g}_s (y) + C \lambda^5 s^{- 3} \| \pi_s y\|_4^4\). Taking \(A \geqslant 2 C_1\) gives (32.8).\(\Box\)

The flows. Fix \(x\) and \(q\). Let \(\bar{\Phi}\) solve \(\hspace{0.17em} \mathrm{d} \bar{\Phi}_s = - C_s^2 D \bar{V}^{\pm}_s (\bar{\Phi}_s) \hspace{0.17em} \mathrm{d} s + C_s \hspace{0.17em} \mathrm{d} W_s\), \(\bar{\Phi}_q = x\). The drift is \(K_s\)-Lipschitz at fixed \(N\) with linear growth, as \(\|Dy\|\) is bounded. Its coordinate \(Y_s = y_s^{\pm} (\bar{\Phi}_s)\) satisfies \(\hspace{0.17em} \mathrm{d} Y = b^{\pm} (Y) \hspace{0.17em} \mathrm{d} s + H^{\pm} (Y) \hspace{0.17em} \mathrm{d} W\), by the computation of Theorem 10.6 with no error. No \(Z\) appears.

Proof. Let \(\xi = Y - y\). Then \(| \pi_s Y - \pi_s y| \leqslant \min (| \xi |, 2 r_s)\), so \(\| \pi_s Y\|_4^4 \leqslant 8 \| \pi_s y\|_4^4 + 32 r_s^2 \| \xi \|_2^2\), and \(\| \pi_s Y\|_2^2 \leqslant 2 \|y\|_2^2 + 2 \| \xi \|_2^2\). So the exponent is at most a deterministic term plus \(\int \kappa_s w_s \hspace{0.17em} \mathrm{d} s\), with \(w = \| \xi \|_2^2\) and \(\kappa_s = C \lambda^4 s^{- 2}\). By Lemma 32.8, \(\text{diag} (H^{\pm})^2 \leqslant Cs^{- 1 / 2}\) and \(\|H^{\pm} \| \leqslant (1 + \varepsilon_s) / s\):

Let \(\theta_s = 4 C \lambda^4 / s\). For \(s\) large, \(\theta' + C \lambda s^{- 3 / 2} \theta + \frac{5}{2} s^{- 2} \theta^2 + \kappa \leqslant 0\), because \(\theta' = - 4 \kappa\) dominates. Then \(\exp \left( \theta_s w_s + \int_q^s \kappa w - \int_q^s a \right)\), with \(a_s = \theta_s (C \lambda s^{- 3 / 2} \|y\|_2^2 + Cs^{- 1 / 2})\), is a local supermartingale by Itô's formula. It starts from \(1\) as \(w_q = 0\), and \(\theta w \geqslant 0\). Fatou gives \(\bar{\mathbb{E}} \exp \left( \int_q^{\infty} \kappa w \right) \leqslant \exp \int_q^{\infty} a\), and \(\int_q^{\infty} a \leqslant C \lambda^5 q^{- 3 / 2} \|y\|_2^2 + C \lambda^4 q^{- 1 / 2}\). The deterministic term is at most \(C \lambda^5 \int_q^{\infty} s^{- 3} \|y\|_4^4 \hspace{0.17em} \mathrm{d} s + C \lambda^4 \int_q^{\infty} s^{- 2} \|y\|_2^2 \hspace{0.17em} \mathrm{d} s\).\(\Box\)

Proof. Let \(\xi_s\) be the free flow, \(\hspace{0.17em} \mathrm{d} \xi = C_s \hspace{0.17em} \mathrm{d} W\), \(\xi_q = x\). Then \(\mathrm{e}^{- V_q (x)} =\mathbb{E} \mathrm{e}^{- V_T (\xi_T)}\) for \(T > q\), as \(V\) solves the Polchinski equation. By Lemma 32.7 and Itô's formula, \(\exp \left( - \bar{V}_T^{\pm} (\xi_T) + \bar{V}_q^{\pm} (x) + \int_q^T (e^{\pm} \circ y + c) \right)\) is the Girsanov density of the flow \(\bar{\Phi}\) with respect to \(\xi\) on \([q, T]\); it is a true martingale, because the drift has linear growth. Therefore, with \(\Delta^{\pm}_T = V_T - \bar{V}_T^{\pm}\),

At fixed \(N\): \(|c (s) | \leqslant C_N s^{- 2}\) by \(H_s (0) - C_s = O (C_s M_s C_s)\) and \(a_2 = O_N (s^{- 2})\), so \(\mathcal{C}_N\) is finite.

Upper bound for \(V_q\). Use \(\Gamma^+\) and Jensen in (32.11): \(V_q (x) \leqslant \bar{V}_q^+ (x) + \int_q^T c + \bar{\mathbb{E}}^+ \Delta^+_T (\bar{\Phi}_T) + \bar{\mathbb{E}}^+ \int_q^T e^+\). By (32.8) and Jensen again, the last term is bounded by the right side of (32.9). As \(\delta \geqslant 0\), \(\Delta^+_T \leqslant V_T - \bar{V}_T\). This tends to \(0\) uniformly on \(\|x\|_{\infty} \leqslant T^{1 / 8}\), by the estimates of the next paragraph. Outside this set it has polynomial domination uniform in \(T\): \(V_T \leqslant C_N (1 +\|x\|_{\infty}^4)\) by Jensen, and \(\bar{V}_T \geqslant - C_N\) since \(U\) dominates \(\tilde{h}_T\). And \(\bar{\Phi}_T\) has all moments uniformly in \(T\) at fixed \(N\), by the linear bound on \(b^+\) and \(\int^{\infty} \text{Tr} (H^+)^2 \hspace{0.17em} \mathrm{d} s < \infty\) at fixed \(N\). So the limsup of \(\bar{\mathbb{E}}^+ \Delta^+_T\) is at most \(0\).

Lower bound. Let \(\mathcal{G}_T\) be the event \(\| \xi_T \|_{\infty} \leqslant T^{1 / 8}\). Since \(V_T \geqslant \inf V_N > - \infty\) at fixed \(N\), \(\mathbb{E} \mathrm{e}^{- V_T (\xi_T)} 1_{\mathcal{G}_T^c} \leqslant C_N \mathbb{P} (\mathcal{G}_T^c) \to 0\). On \(\mathcal{G}_T\), \(\| \xi_T \|_{\infty} \ll r_T\), so \(\delta_T\) vanishes at the coordinate and \(\bar{V}^-_T = \bar{V}_T\). We claim \(\sup_{\mathcal{G}_T} | \Delta^-_T | \to 0\). Indeed at fixed \(N\): \(| \mu_T - \mu_N | = O_N (T^{- 1})\); \(\| \Theta_T \|_{\text{row}} = O_N (T^{- 1 / 2})\); \(\textsf{S}_4 (h_T) = O_N (T^{- 2})\), by the integrated bound using \(a_2 (T) = O_N (T^{- 2})\); and \(|y - x| \leqslant C_N T^{- 1} (\|x\|_{\infty} +\|x\|_{\infty}^3)\). These give \(| \bar{V}_T (x) - V_N (x) | = O_N (T^{- 1 / 4})\) for \(\|x\|_{\infty} \leqslant T^{1 / 8}\). Further \(|V_T (x) - V_N (x) | = O_N (T^{- 1 / 8})\) on the same set, by Gaussian integration, since the remaining variance per site is \(O_N (T^{- 1})\). So

By (32.8) and (32.9) the expectation is at most the exponential of the right side of (32.9), and \(T \to \infty\).

Conclusion. If \(\|y\|_{\infty} \leqslant \frac{15}{16} r_q\), then \(\delta_q\) and \(D \delta_q\) vanish at \(y\). So \(y\) is the critical point of the strictly convex functional defining \(\bar{V}^{\pm}_q (x)\), and \(\bar{V}^{\pm}_q (x) = \bar{V}_q (x)\).\(\Box\)

32.6Uniqueness through the comparison

Proof. Fix \(\delta \in \left( 0, \frac{1}{12} \right)\), \(\rho_q = q^{5 / 12 + \delta}\), and \(\chi_q (x) = \tau (\|x\|_{\infty} / \rho_q)\) with \(\tau\) continuous, \(\tau = 1\) on \(\left[ 0, \frac{1}{2} \right]\), \(\tau = 0\) on \([1, \infty)\).

Step 1: the good set. By (12.5), \(\Phi_q = Y_q + \varrho_q (Y_q) + \frac{\lambda}{3} C_q Y_q^3\), so \(\| \Phi_q \|_{\infty} \leqslant 2 \|Y_q \|_{\infty} + \frac{\lambda}{3 q} \|Y_q \|_{\infty}^3\). Proposition 14.4, whose constant involves \(\|C_{q_0} \|_{2 \to \infty}\), finite with \(a_2 (q_0) \leqslant q_0^{- 2} + C\) uniformly in \(N\), with Lemma 14.5 and Theorem 16.2 gives Lemma 17.1: \(\|Y_q \|_{\infty} \leqslant \mathcal{Y}q^{5 / 12} \log^{1 / 6} (2 + q)\) for \(q \geqslant q_{\ast}\), with \(\mathcal{Y}\) having moments uniformly in \(N\). Therefore \(p_q : = \sup_N \mathbb{P} (\| \Phi^N_q \|_{\infty} > \rho_q / 2) \to 0\).

Step 2: the error. If \(\|x\|_{\infty} \leqslant \rho_q\) then \(\|y_q (x)\|_{\infty} \leqslant 3 \rho_q\). Indeed with \(u = y + \varrho_q (y)\), \(K_q (u - x) + \frac{\lambda}{3} y^3 = 0\). At a positive maximum of \(u - x\), \(K_q (u - x) > 0\), so \(y < 0\) there, and \(u - x \leqslant \| \varrho_q (y)\|_{\infty} + \|x\|_{\infty}\); similarly at a negative minimum. So \(\|u\|_{\infty} \leqslant 2 \|x\|_{\infty} + \varepsilon_q \|y\|_{\infty}\), and \(\|y\|_{\infty} \leqslant 3 \|x\|_{\infty}\) for \(\varepsilon_q\) small. For \(q\) large, \(3 \rho_q \leqslant \frac{15}{16} r_q\), and (32.10) gives on \(\text{supp} \chi_q\)

the last term coming from \(C \lambda^4 q^{- 1} \|y\|_2^2\) with \(\|y\|_{\infty} \leqslant 3 \rho_q\); it tends to zero as \(\delta < \frac{1}{12}\).

Step 3: total variation. \(\Phi^N_q\) has law \(\mu_{N, q} \propto \mathrm{e}^{- V_{q, N}} \gamma_{Q_q}\), with \(Q_q = C - C_q\). Let \(\hat{\nu}_{N, q} \propto \chi_q \mathrm{e}^{- \bar{V}_{q, N}} \gamma_{Q_q}\). On \(\text{supp} \chi_q\) the two unnormalised densities have ratio \(\mathrm{e}^{-\mathcal{C}_N (q) + \theta}\), \(| \theta | \leqslant \varepsilon_q'\); so the density of \(\hat{\nu}_{N, q}\) with respect to the measure \(\chi_q \mu_{N, q} / \int \chi_q \hspace{0.17em} \mathrm{d} \mu_{N, q}\) lies in \([\mathrm{e}^{- 2 \varepsilon_q'}, \mathrm{e}^{2 \varepsilon_q'}]\). The constant cancels, whether or not it converges. Removing \(\chi_q\) costs at most \(2 p_q\) in total variation, so

uniformly in \(N\), the convention being the total mass of the difference.

Step 4: the reference law converges at fixed \(q\). Let \(x_N \sim \gamma_{Q_{q, N}}\) be coupled through common Fourier Gaussians. The multiplier \((q - q_0) ((q_0 + c_k) (q + c_k))^{- 1} \lesssim (1 + |k|)^{- 4}\) gives, by the Kolmogorov criterion, \(\iota_N x_N \to x\) in probability in \(L^{\infty}\). Let \(y = y_q (x)\) and \(w = y - x = - \varrho_q (y) - \frac{\lambda}{3} C_q y^3\). Then

On \(\text{supp} \chi_q\), \(\|y\|_{\infty} \leqslant 3 \rho_q\). Since \(| \hat{\Theta}_q (k) | \leqslant 2 |k| v_1 (\Theta_q)\) we have \(\|C_q \Theta_q \|_{L^2 \to H^1} \leqslant Cv_1\), and the other terms in \(w\) are in \(H^2\), so \(\|w\|_{H^1} \leqslant C_q\) uniformly in \(N\). So the terms involving \(\Theta\) split as

all bounded on the support except the second chaos \(\langle x, \Theta_q x \rangle\). Its operator \(T = Q_q^{1 / 2} \Theta_q Q_q^{1 / 2}\) has multiplier at most \(\operatorname{Cmin} (|k|^{- 3}, \lambda^2 q^{- 1 / 2} q_0^{- 1 / 2})\), so it is Hilbert–Schmidt, with \(\|T\|_{\text{op}} \leqslant \frac{1}{4}\) for \(q\) large. Therefore the centred form converges in \(L^2\) by Proposition 22.1 and dominated convergence in the multipliers, and its exponential is uniformly integrable, by the determinant formula. The trace is a constant, which cancels on normalising. We need that \(D \tilde{h}^N_q\) converges weakly along strongly convergent bounded fields at fixed \(q\). For smooth bounded profiles this is in Proposition 29.1, the finite recursion being uniform on compact sets of scales; it extends to strongly convergent bounded fields by the Lipschitz constant \(\textsf{S}_2 \leqslant Cg_q q\) on \(L^2\). The same holds for the term \(M_q y\) by the fixed \(k\) convergence of Proposition 22.1. With this, \(L^2\) compactness of \(w_N\) from the \(H^1\) bound passes the coordinate equation to the limit. Its solution is unique among bounded fields \(y\) with \(y - x \in H^1\): if \(d\) is the difference of two solutions \(y\), \(y'\), test \(K_q d + \frac{\lambda}{3} (y^3 - (y')^3) + M_q d + D \tilde{h}_q (y) - D \tilde{h}_q (y') = 0\) with \(d\), and use \(\langle d, M_q d \rangle \geqslant - \varepsilon_q \langle d, K_q d \rangle\) and \(\|D \tilde{h}_q (y) - D \tilde{h}_q (y')\|_2 \leqslant Cg_q q \|d\|_2\), both of which pass from the lattice; so \(d = 0\). Thus \(\iota_N y_N \to y\) in \(L^2\) and boundedly. Hence \(U (y_N)\), \(\|y_N \|_2^2\) and \(\tilde{h}_q^N (y_N) = \int_0^1 \langle D \tilde{h}^N_q (ty_N), y_N \rangle \hspace{0.17em} \mathrm{d} t\) converge; so do the cross terms, pairing strong \(x_N\) with weak \(\Theta^N w_N\). So \(\chi_q \mathrm{e}^{- \bar{V}_{q, N} + \text{const}_N}\) converges in probability and in \(L^1 (\lambda)\). The normalisers are bounded below: since \(\bar{V}_q (x) \leqslant \bar{\Gamma}_q (x)\) by taking \(y' = x\) in the infimum, the density is bounded below on the intersection of a small uniform ball with the event that the centred chaos is at most \(K\), and this event has positive mass, uniformly in \(N\), for \(K\) large. Thus \(\iota_N \hat{\nu}_{N, q}\) converges weakly to a limit \(\hat{\nu}_q\), along the whole sequence.

Step 5: from the scale to the field. For smooth \(h\), as \(\Phi_q = J_q \Phi_{\infty} + C_q B_{q - q_0}\) and \(I - J_q = C_q K_{q_0}\),

By (17.6), the first has mean absolute value at most \(C \|C_q K_{q_0} j_N h\|_{\ast} \leqslant Cq^{- 1} \|h\|_{C^2}\), as \(C_q\) has norm \(q^{- 1}\) on each \(L^p\). The second has variance at most \(\|h\|_2^2 / q\). So for \(G = g (\langle \cdot, h_1 \rangle, \ldots, \langle \cdot, h_k \rangle)\) with \(g\) bounded and Lipschitz,

Let \(q \to \infty\). Cylinder functions separate measures on \(H^{- s}\) and Theorem 17.5 gives tightness, so the whole sequence converges. Since \(B^N\) is a Brownian motion independent of \(\Phi^N_{\infty}\), independence passes to the limit, with \(\Phi_q = (q - q_0) C_q \Phi + C_q B_{q - q_0}\).\(\Box\)

What this proof does not give. This argument identifies only the joint law of \(\Phi\) and the Polchinski process. It does not by itself give uniqueness for the system of Theorem 32.5(b) as an equation, nor that \(\psi_q\) is measurable for the filtration of the Polchinski process; these follow from the density in Lecture 7, Theorem 18.6(b) and Theorem 18.11. It does not cover \(H^{- s}\) with \(s\) near \(\frac{1}{2}\) for quartic moments.

32.7Symmetrised Girsanov comparison

This section gives a second proof of Steps 2–3 of Theorem 32.13, which replaces all of Section 32.5. It follows the idea of Gubinelli and Meyer for sine–Gordon [GM24], Proposition 4.9, where the limit law is written as an explicit density over an explicit SDE, and the backward remainder disappears from the identification. There the smoothed force \(Q_t F_t\) and the defect are bounded uniformly in the field [GM24], Proposition 3.13, there is a spatial cut-off, and the flow starts from zero. Here the defect is unbounded, and this older route uses the action only above \(q_{\ast}\), without the frozen extension of Section 12.4, so we use the symmetrised entropy at a fixed scale, which needs only expectations of the defect, and let the scale tend to infinity at the end. In particular neither the penalty, nor the sub- and supersolutions, nor the exponential Riccati bound, nor Lemma 32.9 is needed.

Let \(y = y_q (x)\) solve \(K_q y + D \bar{\Gamma}_q (y) = K_q x\), and let \(\bar{V}_q (x) = \bar{\Gamma}_q (y) + \frac{1}{2} \langle x - y, K_q (x - y) \rangle\) be the Moreau transform of the action. Let \(e_q = \partial_q \bar{\Gamma}_q - \frac{1}{2} \text{Tr} [H_q - H_q (0)]\) be its defect, and let \(V_q\) be the exact potential, \(\mathrm{e}^{- V_q (x)} =\mathbb{E} \mathrm{e}^{- V_N \left( x + \int_q^{\infty} C_s \hspace{0.17em} \mathrm{d} W_s \right)}\), with force \(F_q = - DV_q\). By the envelope theorem and \(Dy = H_q K_q\),

So \(\bar{V}\) solves the Polchinski equation up to its defect, and \(\hspace{0.17em} \mathrm{d} \bar{\Phi} = - C_s^2 D \bar{V}_s (\bar{\Phi}) \hspace{0.17em} \mathrm{d} s + C_s \hspace{0.17em} \mathrm{d} W\) is an explicit flow, whose coordinate \(Y = y_s (\bar{\Phi}_s)\) solves \(\hspace{0.17em} \mathrm{d} Y = b (Y) \hspace{0.17em} \mathrm{d} s + H (Y) \hspace{0.17em} \mathrm{d} W\), \(b = - HDe\): indeed \(\hspace{0.17em} \mathrm{d} (K_s \bar{\Phi}) = Y \hspace{0.17em} \mathrm{d} s + \hspace{0.17em} \mathrm{d} W\), and one applies Itô's formula to \(K_s Y + D \bar{\Gamma}_s (Y) = K_s \bar{\Phi}\). It contains no \(Z\).

Proof. On the ball \(e_s = e_s^{\text{o}}\), so \(|e_s (\pi_s y) | \leqslant C \lambda^5 s^{- 3} \| \pi_s y\|_4^4 + C \lambda^4 s^{- 2} \| \pi_s y\|_2^2\) by the zero scalar jets, the centred quadratic jet and (11.14). With \(z_{\theta} = \pi_s y + \theta (y - \pi_s y)\), \(e_s (y) - e_s (\pi_s y) = \int_0^1 \langle De_s (z_{\theta}), y - \pi_s y \rangle \hspace{0.17em} \mathrm{d} \theta\), and \(y - \pi_s y\) is supported on the set \(\mathcal{A}\) where \(|y| > r_s\), with norm at most \(\|t_s \|_2\). Write \(De = D \kappa De^{\text{o}} \circ \kappa +\mathfrak{D}\). Then \(|De^{\text{o}} | \leqslant \textsf{S}_4 (e^{\text{o}}) (2 r_s)^3 / 6 + \|\mathcal{E}^c_s \|_{\text{row}} 2 r_s \leqslant C \lambda^{7 / 2} s^{- 3 / 2}\) and \(|\mathcal{A}|^{1 / 2} \leqslant r_s^{- 1} \|t_s \|_2\), so the first term contributes at most \(C \lambda^4 s^{- 2} \|t_s \|_2^2\). By (21.11), which bounds \(\mathfrak{D}\) before multiplication by \(H\) (the bound stated for \(b^{\text{l}} = - H\mathfrak{D}\) would not suffice, as \(H^{- 1}\) is unbounded), \(\|\mathfrak{D}_s (z)\|_2 \leqslant Cg_s (\|t_s (z)\|_2 + r_s^{- 1} \|z^2 \chi_s (z)\|_2)\); since \(|z_{\theta} | \leqslant |y|\) and \(z_{\theta}\) has size above \(r_s\) only on \(\mathcal{A}\), the second term contributes at most \(Cg_s (\|t_s \|_2^2 + r_s^{- 1} \|t_s \|_4^2 \|t_s \|_2)\). On the support of \(t_s\) we have \(|t_s | \geqslant r_s\), so \(\|t_s \|_2 \leqslant r_s^{- 2} \|t_s \|_6^3\) and \(\|t_s \|_4^2 \leqslant r_s^{- 1} \|t_s \|_6^3\). All these are at most \(Cg_s r_s^{- 4} \|t_s \|_6^6 = C \lambda^3 s^{- 5 / 2} \|t_s \|_6^6\).\(\Box\)

Fix \(\delta \in \left( 0, \frac{1}{12} \right)\), \(\rho_q = q^{5 / 12 + \delta}\), and \(\chi_q (x) = \tau (\|x\|_{\infty} / \rho_q)\), with \(\tau\) continuous, \(\tau = 1\) on \(\left[ 0, \frac{1}{2} \right]\), \(\tau = 0\) on \([1, \infty)\). Let \(\mu_{N, q} \propto \mathrm{e}^{- V_{q, N}} \gamma_{Q_q}\) be the law of \(\Phi_q^N\), and \(\hat{\nu}_{N, q} \propto \chi_q \mathrm{e}^{- \bar{V}_{q, N}} \gamma_{Q_q}\) the explicit law.

Proof. The two path laws. On \([q, \infty)\) let \(\mathbb{P}\) be the exact law, conditioned softly: initial law \(\chi_q \mu_{N, q} / \int \chi_q \hspace{0.17em} \mathrm{d} \mu_{N, q}\), drift \(C_s^2 F_s\). Let \(\mathbb{R}\) be the explicit one: initial law \(\hat{\nu}_{N, q}\), drift \(- C_s^2 D \bar{V}_s\). Both are absolutely continuous with respect to the free flow \(\hspace{0.17em} \mathrm{d} \xi = C_s \hspace{0.17em} \mathrm{d} W\) with initial law \(\gamma_{Q_q}\). On \([q, T]\) the first has density proportional to \(\chi_q (\xi_q) \mathrm{e}^{- V_T (\xi_T)}\), since \(\mathrm{e}^{- V}\) is harmonic. By (32.13) and Itô's formula, the second has density proportional to

This is a true martingale density: the drift is \(- C_s (x - y_s (x))\), Lipschitz in \(x\) at fixed \(N\) since \(Dy = HK\). The factors at \(\xi_q\) cancel: this is why no estimate at the initial scale is needed. So \(\log \hspace{0.17em} \mathrm{d} \mathbb{P}/ \hspace{0.17em} \mathrm{d} \mathbb{R}= - \Delta_T (\xi_T) - \int_q^T e_s (Y_s) \hspace{0.17em} \mathrm{d} s + \text{const}\), with \(\Delta_T = V_T - \bar{V}_T\). The constant includes \(\int c\) and the normalisations, which diverge with \(N\); it does not matter, since in the symmetrised relative entropy it cancels:

At fixed \(N\): \(| \mu_T - \mu_N | = O_N (T^{- 1})\), \(\| \Theta_T \|_{\text{row}} = O_N (T^{- 1 / 2})\), \(\textsf{S}_4 (h_T) = O_N (T^{- 2})\) and \(|y - x| \leqslant C_N T^{- 1} (\|x\|_{\infty} +\|x\|_{\infty}^3)\) give \(| \bar{V}_T - V_N | = O_N (T^{- 1 / 4})\) on \(\|x\|_{\infty} \leqslant T^{1 / 8}\), and Gaussian integration, the remaining variance per site being \(O_N (T^{- 1})\), gives \(|V_T - V_N | = O_N (T^{- 1 / 8})\) there. For all \(x\), \(| \Delta_T (x) | \leqslant C_N (1 +\|x\|_{\infty}^4)\) uniformly in \(T\): \(\inf V_N \leqslant V_T (x) \leqslant C_N (1 +\|x\|_{\infty}^4)\) by Jensen, and \(- C_N \leqslant \bar{V}_T (x) \leqslant \bar{\Gamma}_T (x) \leqslant C_N (1 +\|x\|_{\infty}^4)\), as \(U\) dominates \(\tilde{h}_T\). The moments of \(\xi_T\) are uniform in \(T\): under \(\mathbb{P}\) since \(\Phi_T = J_T \Phi_{\infty} + C_T B_{T - q_0}\); under \(\mathbb{R}\) by the Ito computation below with \(\int^{\infty} a_2 \hspace{0.17em} \mathrm{d} s < \infty\) at fixed \(N\), and \(x = Y + \varrho_T (Y) + \frac{\lambda}{3} C_T Y^3\). So \(\mathbb{E}^{\mathbb{P}} \Delta_T, \mathbb{E}^{\mathbb{R}} \Delta_T \to 0\). With the bounds below, \(\log \hspace{0.17em} \mathrm{d} \mathbb{P}/ \hspace{0.17em} \mathrm{d} \mathbb{R}\) is integrable under both laws at each \(T\), so both entropies are finite and (32.15) is justified. By Pinsker's inequality the squared total variation at the initial scale is at most the sum of the two expectations of \(\int_q^{\infty} |e_s (Y_s) | \hspace{0.17em} \mathrm{d} s\), and removing \(\chi_q\) costs at most \(2 p_q\). It remains to bound these expectations, by (32.14).

Under the exact law. \(\int_q^{\infty} \lambda^3 s^{- 5 / 2} \|t_s \|_6^6 \hspace{0.17em} \mathrm{d} s \leqslant \lambda^2 q^{- 1} J_{\infty}\). Further \(\| \pi_s Y_s \|_4^4 \leqslant r_s^2 \|Y_s \|_2^2\), and \(\mathbb{E} \|Y_s \|_2^2 \leqslant C (1 + s^{1 / 2})\) by Itô's formula for \(\|Y_s \|_2^2\) along the forward equation of Theorem 10.6, from \(q_{\ast}\), with the moments of \(Y_{q_{\ast}}\) and \(\mathbb{E} \|Z_u \|_2^2 \leqslant Cu^{- 5 / 2}\) of Theorem 16.2, \(\|H\| \leqslant C / u\), the linear bound on \(b\) and \(\text{Tr} H^2 \leqslant Cu^{- 1 / 2}\). So the first two terms of (32.14), both at most \(C \lambda^4 s^{- 2} \|Y_s \|_2^2\), integrate in expectation to at most \(C \lambda^4 q^{- 1 / 2}\). The conditioning costs at most a factor \(2\).

Under the explicit law. Here \(Y_q = y\) with \(\|y\|_{\infty} \leqslant 3 \rho_q\) by the maximum principle: with \(u = y + \varrho_q (y)\), \(K_q (u - x) + \frac{\lambda}{3} y^3 = 0\), and at a positive maximum of \(u - x\) we have \(y < 0\), so \(u - x \leqslant \| \varrho_q (y)\|_{\infty} + \|x\|_{\infty}\); similarly at a negative minimum. Thus \(\|u\|_{\infty} \leqslant 2 \|x\|_{\infty} + \varepsilon_q \|y\|_{\infty}\), with \(\varepsilon_q\) the contraction constant of (12.5), and \(\|y\|_{\infty} \leqslant 3 \|x\|_{\infty}\) for \(q\) large. Let \(\xi_s = Y_s - y\), which is a different \(\xi\) from the free flow. By the global linear bound of Proposition 12.2, \(\|b (Y)\|_p \leqslant C \lambda s^{- 3 / 2} \|Y\|_p\) for every \(p\) (it follows from the exact gradient of \(e\), without the ledger), and \(\text{diag} H^2 \leqslant Cs^{- 1 / 2}\), Itô's formula gives, for \(p = 2\) and \(p = 12\), \(\mathbb{E} \| \xi_s \|_p^p \leqslant C (s^{p / 4} + (g_q \rho_q)^p)\). So, as \(\| \pi_s Y\|_4^4 \leqslant 8 \|y\|_4^4 + 32 r_s^2 \| \xi \|_2^2\), and \(\| \pi_s Y\|_2^2 \leqslant 2 \|y\|_2^2 + 2 \| \xi \|_2^2\), the first two terms of (32.14) integrate to at most \(C \lambda^5 q^{- 2} \rho_q^4 + C \lambda^4 q^{- 1} \rho_q^2 + C \lambda^4 q^{- 1 / 2}\). For large \(q\), \(r_s \geqslant r_q \geqslant 6 \rho_q\), so where \(|Y| > r_s\) we have \(| \xi | > r_s / 2\) and \(|Y| \leqslant 2 | \xi |\). Thus \(\|t_s \|_6^6 \leqslant Cr_s^{- 6} \| \xi_s \|_{12}^{12}\), and the third term integrates to at most \(C \lambda^6 q^{- 3 / 2}\).\(\Box\)

The bound on \(\mathbb{E}^{\mathbb{P}}\) is where UV stability enters. So the backward equation is used only through Theorem 16.2, for \(J_{\infty}\) and \(\mathbb{E} \|Z_s \|_2^2\); under \(\mathbb{R}\) the comparison has no \(Z\), no conditional expectation and no exponential moment. What is lost is the pointwise bound on \(V_q - \bar{V}_q\) on typical fields which a Feynman–Kac comparison gives: here one gets total variation at a fixed scale, and that is all the theorem needs.

Remark 32.16. (The backward error alone) The drifts differ by \(C_s^2 (F_s - \bar{f}_s)\) with \(\bar{f} = - D \bar{V}\), and \(C_s (F_s - \bar{f}_s) = Z_s\). So, from the same initial law at scale \(q\), Girsanov's theorem gives directly that the relative entropy of the exact path law with respect to the explicit flow is \(\frac{1}{2} \int_q^{\infty} \mathbb{E} \|Z_s \|_2^2 \hspace{0.17em} \mathrm{d} s \leqslant Cq^{- 3 / 2}\), hence total variation \(\leqslant Cq^{- 3 / 4}\). This replaces the backward error by a change of measure, but it does not identify the limit: it reduces the terminal law to the law at scale \(q\), which is the unknown, since the flow does not forget the low modes. In [GM24] this does not arise, as the flow starts from a deterministic point; the same holds in Lecture 7, where the frozen action starts the flow at \(q_0\) from zero (Theorem 18.5). The symmetrised entropy above compares the laws at scale \(q\) as well, through the cancellation of the factors at \(\xi_q\).

With Proposition 32.15, Steps 1, 4 and 5 of Theorem 32.13 conclude the proof as before: it uses Theorem 16.2, Proposition 14.4, Lemma 14.5, (11.14), Lemma 12.1, and Proposition 12.2 with its ledger. On this route the condition \(q_{\ast} \geqslant q_A\) of Theorem 32.13 is not needed.

References

[GM24]

M. Gubinelli, S.-J. Meyer. The FBSDE approach to sine–Gordon up to \(6 \pi\). Preprint, arXiv:2401.13648, 2024.