A Ray-Knight theorem for $\nabla\phi$ interface models and scaling limits

We introduce a natural measure on bi-infinite random walk trajectories evolving in a time-dependent environment driven by the Langevin dynamics associated to a gradient Gibbs measure with convex potential. We derive an identity relating the occupation times of the Poissonian cloud induced by this measure to the square of the corresponding gradient field, which - generically - is not Gaussian. In the quadratic case, we recover a well-known generalization of the second Ray-Knight theorem. We further determine the scaling limits of the various objects involved in dimension 3, which are seen to exhibit homogenization. In particular, we prove that the renormalized square of the gradient field converges under appropriate rescaling to the Wick-ordered square of a Gaussian free field on $\mathbb{R}^3$ with suitable diffusion matrix, thus extending a celebrated result of Naddaf and Spencer regarding the scaling limit of the field itself.


Introduction
Random-walk representations and isomorphism theorems have a long history in mathematical physics and probability theory, going back at least to works of Symanzik [68], Ray [60] and Knight [45], among others; we refer to the monographs [35,54,49,72] and references therein for a more exhaustive overview.Recent developments, not captured by these references, include signed versions of some of these identities and their characterization through cluster capacity observables, see [50,52,74,30], continuous extensions in dimension two [9,3], applications to percolation problems in higher dimensions [50,29], to cover times, see e.g.[26,27,41,1], and generalizations to different target spaces [10,11,42,51], with ensuing relevance e.g. to the study of reinforced processes.
In the present article, we investigate similar questions for a broader class of (generically) non-Gaussian scalar gradient models introduced by Brascamp, Lebowitz and Lieb in [18], which have received considerable attention, see [19,58,36,38,8] and further references below, in particular, Remark 5.2, (2).In a sense, our findings assess the "stability" of such identities under gradient perturbations.
We now explain our main results, which appear in Theorems 4.3 and 5.1 below.We consider the lattice Z d , for d ě 3, and for ϕ : Z d Ñ R the (formal) Hamiltonian where the sum ranges over x, y P Z d and | ¨| denotes the Euclidean norm.We will assume for simplicity (but see Remark 7.6,(3) below with regards to relaxing the assumptions on U ) that U is even, U P C 2,α pRq, for some α ą 0 and for some c 1 , c 2 P p0, 8q.We write E " R Z d , endowed with the corresponding product σ-algebra F and corresponding canonical coordinate maps ϕ x : E Ñ R for x P Z d .We then consider, for finite Λ Ă Z d and all ξ P E, the probability measure on pE, Fq defined as where H Λ is obtained from H by restricting the summation in (1.1) to (neighboring) vertices x, y such that tx, yu X Λ ‰ H.The condition (1.2) guarantees in particular that (1.3) is well-defined.
Associated to this setup is a Gibbs measure µ on pE, Fq, defined as the weak limit (1.4) µ def. " where µ per Λ N ,ε refers to the analogue of the finite-volume measure in (1.3) with Λ N " pZ{2 N Zq d (periodic boundary conditions) and with H Λ pϕq replaced by the massive Hamiltonian H Λ pϕq ὲ x , ε ą 0. Indeed combining the Brascamp-Lieb inequality [17,18] and the bounds of [23], one classically knows that the measures µ per Λ N ,ε are tight in N and their subsequential limits tight in ε, hence the limits in (1.4) exist, possibly upon possibly passing to appropriate subsequences.The Gibbs property of µ is the fact that, for any finite set Λ Ă Z d , with F Λ " σpϕ x : x P Z d zΛq, µp ¨|F Z d zΛ qpξq " µ ξ Λ p¨q, µpdξq-a.s.(1.5)The measure µ will be the main object of interest in this article.We use E µ r¨s to denote expectation with respect to µ in the sequel.By construction, µ is translation-invariant, ergodic with respect to the canonical lattice shifts τ x : E Ñ E, x P Z d , and E µ rϕ x s " 0 for all x P Z d .
As will turn out, our scaling limit results require probing squares of the canonical field ϕ under µ in a sequence of growing finite subsets exhausting Z d , thus leading to generating functionals that involve tilting the measure µ by both linear and quadratic functionals of the field.We now introduce these measures, which are parametrized by a function h and a (typically) signed potential V , with corresponding Hamiltonian (cf.(1.1)) (1.6) H h,V pϕq def.
" Hpϕq x (the minus signs are a matter of convenience), where h, V : Z d Ñ R have finite support and }V `}8 ¨diampsupppV `qq 2 ă λ 0 .(1.7) Here, λ 0 " cpd, c 1 q P p0, 8q, where V `" maxtV, 0u is the positive part of V , supppV q " tx P Z d : V pxq ‰ 0u and diam refers to the ℓ 8 -diameter of a set; see Remark 2.4,(2) below regarding the choice of λ 0 .Under (1.7), we introduce the probability measure µ h,V on pE, Fq defined by (note in particular that µ " µ 0,0 ); we refer to Lemma 2.3 and Remark 2.4 for matters relating to the tilt in (1.8) under condition (1.7), which, along with (1.2), we always assume to be in force from here on.The measure µ h,V is a Gibbs measure for the specification pU, h, V q.In case h " 0, µ 0,V is invariant under ϕ Þ Ñ ´ϕ and has zero mean.Moreover, if U pηq " 1 2 η 2 , then µ h,V is the Gaussian free field on Z d (with 'mass' V when V ď 0 and non-zero mean unless h " 0).
We now introduce certain dynamics corresponding to the above setup, which will play a central role in this article.One naturally associates to µ h,V in (1.8) a diffusion tϕ t : t ě 0u on E attached to the Dirichlet form (1.9) with domain in L 2 pµ h,V q, which is the domain of the closed self-adjoint extension of the second order elliptic operator L 1 in L 2 pµ h,V q, where (1.10) which acts on local (i.e.depending on finitely many coordinates) smooth bounded functions f : E Ñ R such both Bf Bϕx and B 2 f Bϕ 2 x are bounded.The assumptions (1.2),(1.7)ensure that the construction of tϕ t : t ě 0u falls within the realm of standard theory; indeed tϕ t : t ě 0u is obtained as a solution to the system of SDE's (1.11) dϕ t pxq " !´ÿ y: |y´x|"1 U 1 pϕ t pxq ´ϕt pyqq `V pxqϕ t pxq `hpxq ) dt `?2dW t pxq, x P Z d with appropriate initial conditions in tϕ P E : ř x |ϕ x | 2 e ´λ|x| ă 8 for some λ ą 0u, where pW t pxqq xPZ d is a family of independent standard Brownian motions.The relevant drift terms in (1.11) are globally Lipschitz and guarantee the existence of a unique solution for the associated martingale problem [65].
The assumptions (1.7) ensure that the weights (1.12) are uniformly elliptic and the construction of the corresponding Markov chain on Z d is standard.We will be interested in the evolution of the process X t " pX t , ϕ t q on Z d ˆE generated by for suitable f , and the corresponding Dirichlet form with domain DpEq in L 2 pρ h,V q, where ρ h,V " κ ˆµh,V , with κ counting measure on Z d , given by Epf, f q " pf p¨, ϕq, f p¨, ϕqqµ h,V pdϕq. (1.16) Note in particular that L is symmetric with respect to ρ h,V , that is, for suitable f and g, In line with above notation, we abbreviate ρ " ρ 0,0 , whence ρ " κ ˆµ.We write P px,ϕq for the canonical law of X ¨started at px, ϕq.This is a probability measure on the space W `of right-continous trajectories on Z d ˆE whose projection on Z d escapes all finite sets in finite time.
We use θ t , t ě 0, to denote the corresponding time-shift operators.It will often be convenient to write, for f " f pX ¨q bounded and supported on tX 0 P Au, for some finite A Ă Z d , (1.18) E ρ h,V rf s " ÿ x ż E µ h,V pdϕqE px,ϕq rf s ˆ" ż ρ h,V pdx, dϕqE px,ϕq rf s ˙.
The process X ¨is deeply linked to µ h,V .Indeed, adapting the arguments of [24,38], one knows that for all functions F, G : E Ñ R satisfying a suitable growth condition at infinity, comprising in particular any polynomial expression of an arbitrary finite-dimensional marginal of the field ϕ (which will be sufficient for our purposes), (1.19) cov µ h,V pF, Gq " here BF px, ϕq " BF pϕq{Bϕ x , for x P Z d and, with a slight abuse of notation, we regard V as the multplication operator V f px, ϕq " V pxqf px, ϕq, for f : Z d ˆE Ñ R. We refer to [24], Prop.2.2 and Remark 2.3 for a proof of (1.19); see also [39].This formula links covariances associated to the (in general non-Gaussian) random field, ϕ, to a certain Markov process, X.It is thus natural to ask if one has identities resembling the classical isomorphism theorems in the Gaussian case.
Our first result is that this is indeed the case: we derive one such identity in Theorem 4.3 below, which can be regarded as a generalization of the second Ray-Knight theorem.Namely, for a suitable measure P V u which we will introduce momentarily, we prove in Theorem 4.3 that for all u ą 0 and V : Z d Ñ R as in (1.7), with µ as in (1.4), The key here is the measure P V u governing the field L ¨, which we now describe in some detail.In a nutshell, P V u is a Poisson process of trajectories on Z d ˆE modulo time-shift, whose total number is controlled by the scalar parameter u ą 0: the larger u is, the more trajectories enter the picture.The intensity measure ν V u of this process, constructed in Theorem 3.5 below (cf.also (4.10)), is roughly speaking the unique natural measure on such trajectories whose forward part evolve like the process X generated by L as given by (1.15), with a slight twist.Namely, L is not simply the generator for the Langevin dynamics associated to µ " µ 0,0 .Instead, the potential V in (1.20) manifests itself as a drift term in the system of SDE's governing the Langevin dynamics in (1.10),As it turns out, these dynamics are solutions to the SDE's (1.11) where V corresponds exactly to the test function in (1.20) and h is appropriately chosen; see the discussion leading up to (4.2) and (4.10) in Section 4 for precise definitions.
The field L ¨is then simply the cumulated occupation time of the spatial parts of all trajectories in the soup.In case U in (1.6) is quadratic, the components of X decouple, the projection of the process P V u onto the first coordinate has the law of random interlacements and (1.20) specializes to the isomorphism theorem of [71]; see Remarks 3.6 and 4.2 below for details.In particular, the construction of the measure P V u described above entails the interlacement process introduced in [70] as a special case.
The derivation in Theorem 3.5 of the intensity measure lurking behind L ¨in (1.20) involves a patching of several local 'charts' (much like the DLR-condition, see Remark 3.4) and relies on elements of potential theory associated to the process X, see Section 3. The two crucial inputs to do the patching are i) a suitable probabilistic representation of the equilibrium measure for space-like cylinders, and ii) reversibility of X with respect to ρ, which together give rise to a desirable sweeping identity, see Proposition 3.2.Once Theorem 3.5 is shown, the proof of (1.20) in Theorem 4.3 is essentially obtained as a consequence of a suitable Feynman-Kac formula for a killed version of the (big) process X (rather than just X).
We refer to Remark 5.2 below for further comments around isomorphism theorems in the present context of (1.1).We hope to return to applications of (1.20) and other similar formulas, e.g. with regards to existence of mass gaps, elsewhere [25].The utility of identities like (1.20) for problems in statistical mechanics cannot be over-emphasized, where it can for instance be used as a powerful dictionary between the worlds of percolation and random walks in transient setups, see e.g.[61] for early works in this direction, and more recently [50,74,28,75,30,29].We also refer to [62,5] for percolation and first-passage percolation in the context of ∇ϕ-models, as well as the references at the beginning of this introduction for a host of other applications.
A version of our first result, Theorem 4.3, can also be proved on a finite graph with suitable (wired) boundary conditions, see Remark 5.2,(1) below.In case U in (1.6) is quadratic, (1.20) was proved in [34], and later extended to infinite volume in [71] in transient dimensions.We further refer to [63] for a pinned version in dimension 2, to [50,74,59] for a signed version, to [64] for an "inversion", and to [10,11,55,20] for related findings in the context of certain hyperbolic target geometries.Finally, let us mention that, similarly as in the Gaussian case, see [76] or [31,Chap. 3], the Poisson process underlying L in (1.20) can likely be exhibited as the (annealed) local limit of a random walk on the torus defined similarly as the spatial projection of X under P ρ in (3.9); we will not pursue this further here.
Similar in spirit to works of Le Jan [48] and Sznitman [73] in the Gaussian case, we then investigate the existence of possible scaling limits for the various objects attached to (1.20).Our starting point is the celebrated result of Naddaf-Spencer [58] regarding the scaling limit of ϕ itself to a continuum free field Ψ, see (5.5)-(5.6)and (5.13) below (see also Remark 5.2,(2) for related findings among a vast body of work on this topic), whose covariance function is the Green's function of a Brownian motion with homogenized diffusion matrix Σ, obtained as the scaling limit of the first coordinate of X under diffusive scaling, cf.(5.4).Given this homogenization phenomenon for ϕ, (1.20) may plausibly lie in the 'domain of attraction' of a limiting Gaussian identity involving Ψ.
Among other things, our second main result addresses this question.Indeed, we prove in Theorem 5.1 below that, as a random distribution on R 3 , cf.Section 5 for exact definitions, and with ϕ N pzq " N 1{2 ϕ tN zu , z P R 3 , (1.21) p ϕ N , : ϕ 2 N : q under µ converges in law to p Ψ, : Ψ 2 : q as N Ñ 8, (see Theorem 5.1 below for the precise statement), where : " ϕ 2 N p¨q ´Eµ rϕ 2 N p¨qs and : Ψ 2 : stands for the Wick-ordered square of Ψ, see (5.7).Thus, our theorem can be understood as an extension of Naddaf and Spencer's result [58] to the simplest possible non-linear functional of the field, i.e. ϕ 2 , when d " 3.
The nonlinearity in (1.21) is by no means a small issue.The proof of results similar to (1.21) are already delicate in the Gaussian case, see [66,48,73], and even more so presently, due to the combined effects of i) the absence of Gaussian tools, and ii) the need for renormalization.
Our approach also yields a new proof in the Gaussian case, which we believe is more transparent.For instance, it avoids the use of determinantal formulas, such as those typically used to express generating functionals like (1.22) below -in fact our proof yields a different representation of such functionals, see (5.11)-(5.12)and Remark 5.2,(3)).We now briefly outline our strategy and focus our discussion on the marginal : ϕ 2 N : alone in (1.21) for simplicity.We first prove tightness by controlling generating functionals of gradient squares in Proposition 6.1, i.e. for V P C 8 0 pR 3 q and |λ| small enough, we obtain uniform bounds of the form cf. (6.3) below.This is facilitated through the use of a certain variance estimate, see Lemma 2.3 (in particular (2.16)), which is of independent interest and can be viewed as a consequence of the more classical Brascamp-Lieb estimate [17].Once (1.22) is shown, the task is to identify the limit in (1.21).To do so, we first replace ϕ N by a regularized version ϕ ε N , corresponding at the discrete level to the presence of an ultraviolet cut-off in the limit.The removal of the divergence at ε ą 0 allows for an application of [58], which together with tightness estimates akin to (1.22), is seen to imply convergence of pϕ ε N q 2 .To remove the cut-off, the crucial control is the following L 2 -estimate, derived in Section 6.2.Namely, we show in Proposition 6.7 that for all ε ą 0, there exists cpεq P p1, 8q such that (1.23) lim " 0, pd " 3q.
The bound (1.23) is obtained as a consequence of the Brascamp-Lieb inequality alone; no further random walk estimates on X are necessary.In particular, no gradient estimates on its Green's function are needed, as one might naively expect from the form of (1.23) on account of (1.19).
The controls (1.23) are surprisingly strong.For instance, one does not need to tune ε with N when taking limits in (1.21).Rather, one can in a somewhat loose sense first let ε Ñ 0 then N Ñ 8 (cf.Lemmas 7.1-7.3below for precise statements) and (1.23) serves to determine the exact limits of the functionals in (1.22), thus completing the proof.
Returning to the identity (1.20), the result (1.21) then enables us to directly identify the limit of suitably rescaled occupation times L N of L when d " 3, and we deduce in Corollary 7.5 below that L N converges in law to the occupation-time measure of a Brownian interlacement with diffusivity Σ, cf.(7.14)-(7.15)for precise definitions.As in the Gaussian case, the convergence of the associated occupation time measure does not require counter-terms.In particular, the drift term implicit in P V u generated by the potential V , which breaks translation invariance, is thus seen to "disappear" in the limit.Further, we immediately recover from this the limiting isomorphism proved in [73] in the Gaussian case (albeit with non-trivial diffusivity Σ stemming from homogenization), see Corollary 7.7 and (7.23) below.In the parlance of renormalization group theory, (7.23) is thus seen to be the "Gaussian fixed point" of the identity (1.20) for any potential U satisfying (1.2).
We now describe how this article is organized.In Section 2 we gather various useful preliminary results.To avoid disrupting the flow of reading, some proofs are deferred to an appendix (this also applies to several bounds related to ε-smearing in Sections 6-7).In Section 3, we develop some potential theory tools for the process X with generator L, see (1.15), and introduce the intensity measure underlying P V u in (1.20).In Section 4, we state and prove the isomorphism, see Theorem 4.3.Section 5 gives precise meaning to our scaling limit result for the renormalized squares of ϕ.The statement appears in Theorem 5.1 and is proved over the remaining two sections 6-7.Section 6 contains some preparatory work: Sections 6.1 and 6.2 respectively deal with matters relating to tightness (cf.(1.22)) and the aforementioned L 2 -estimate (cf.(1.23)), see also Propositions 6.1 and 6.7 below; Section 6.3 deals with convergence of the smeared field at a suitable functional level.The actual proof of Theorem 5.1 then appears in Section 7, along with its various corollaries, notably the scaling limits of rescaled occupation times (Corollary 7.5) and the limiting isomorphism (Corollary 7.7).
Throughout, c, c 1 , . . .denote positive constants which can change from place to place and may depend implicitly on the dimension d.Numbered constants are fixed upon first appearance in the text.The dependence on any quantity other than d will appear explicitly in our notation.
Acknowledgments.This work was initiated while one of us (JDD) was visiting UCLA, while the other (PFR) was still working there.We both thank Marek Biskup for being the great host he is.PFR thanks TU Berlin for its hospitality on several occasions.We thank M. Slowik for stimulating discussions at the final stages of this project.Part of this research was supported by the ERC grant CriBLaM.We thank two anonymous referees for the quality of their reviews.

Preliminaries and tilting
In this section we first gather several useful results for the discrete Green's function in a potential V .Lemma 2.1 yields useful comparison bounds for the corresponding heat kernel in terms of the standard (i.e. with V " 0) one under suitable assumptions on V .Lemma 2.2 deals with scaling limits of the associated Green's function (and its square).We then discuss key aspects of the ϕ-Gibbs measures µ h,V introduced in (1.8) (see also (1.4)) under the assumptions (1.2),(1.7),including matters relating to existence of µ h,V , which involves exponential tilts with functionals of ϕ 2 ; for later purposes we actually consider general quadratic functionals of ϕ, see (2.12) and conditions (2.13)- (2.14).Some care is needed because the scaling limits performed below will require the tilt to be signed and have finite but arbitrarily large support.We also collect a useful variance estimate, of independent interest, see Lemma 2.3 and in particular (2.16), see also Lemma 2.5 regarding higher moments, which can be viewed as a consequence of the Brascamp-Lieb inequality.
Let pZ t q tě0 denote the continuous-time simple random walk on Z d with generator given by (1.14) with a " 1 (amounting to the choice U ptq " 1 2 t 2 in (1.12)).We write P x for its canonical law with Z 0 " x and E x for the corresponding expectation.For V : Z d Ñ R, we introduce the heat kernels (2.1) q V t px, yq " E x " e ş t 0 V pZsqds 1 tZt"yu ‰ , for x, y P Z d , t ě 0, and abbreviate q t " q 0 t .The corresponding Green's function is defined as (possibly `8) with g 0 " g.We now discuss conditions on V `" maxtV, 0u guaranteeing good control on these quantities, which will be useful on multiple occasions.
Lemma 2.1 (d ě 3).There exists ε ą 0 such that, for any V : and all x, y P Z d , one has: The proof of Lemma 2.1 is deferred to Appendix A. Now, for smooth, compactly supported V : R d Ñ R and arbitrary integer N ě 1, consider its discretization (at level N ) and the rescaled Green's function 2) with V N given by (2.6).In accordance with the notation g " g 0 , cf. below (2.2), we set g N " g 0 N , whence g N pz, for any function f : R d Ñ R such that ş g V N pz, z 1 q k |f pz 1 q|dz 1 ă 8.The operator pG V N q 2 is defined similarly, with kernel g V N pz, z 1 q 2 in place of g V N pz, z 1 q on the right-hand side of (2.8).Finally, we introduce continuous analogues for (2.8).Let W z , z P R d , denote the law of the standard d-dimensional Brownian motion pB t q tě0 starting at z and (2.9) for suitable f , V (to be specified shortly).Let x¨, ¨y refer to the standard inner product on R d .
Lemma 2.2.For all f, V P C 8 0 pR d q with supppV q Ă B L for some L ě 1 and }V } 8 ď cL ´2, In particular (2.10)- (2.11) implicitly entail that all expressions are well-defined and finite, i.e. all of G V N , G V (and pG V N q 2 when d " 3) act on C 8 0 pR d q when the potential V satisfies the above assumptions.The proof of Lemma 2.2 is given in Appendix A.
Next, we introduce suitable tilts of the measure µ defined in (1.4).The ensuing variance estimates below are of independent interest.We state the following bounds at a level of generality tailored to our later purposes.For real numbers Q λ px, yq, x, y P Z d , indexed by λ ą 0 (cf.(2.14) below regarding the role of λ) and vanishing unless x, y belong to a finite set, let (2.12) 2) and abbreviate B x F " BF pϕq{Bϕ x below.The inequality (2.15) below (in the special case where F is a linear combination of ϕ x 's) is due to Brascamp-Lieb, see [17,18].
then e Q λ P L 1 pµq and the following hold: for F P C 1 pE, Rq depending on finitely many coordinates such that F and B x F , x P Z d , are in L 2 pµ Q λ q, one has (2.15) var µ Q λ pF q ď c ÿ x,y gpx, yqE µ Q λ rB x F B y F s.
If moreover, F P C 2 pE, Rq and B x B y F P L 2 pµ Q λ q for all x, y P Z d , then Remark 2.4.( 1) By adapting classical arguments, see e.g.[24, Corollary 2.7], one readily shows that the conclusions of Lemma 2.3 (and thus also of Lemma 2.5 below) continue to hold if one considers the measure µ h,Q λ with exponential tilt of the form Qpϕ, ϕq `řx hpxqϕ x , for arbitrary h as in (1.7).
(2) In particular, Lemma 2.3 applies with the choice (2.17) for V as in (1.7) with λ 0 " c 3 .Indeed with the choice R " diampsupppV qq, one readily finds x 0 such that (2.13) is satisfied.Moreover, with B " Bpx 0 , Rq, (2.17) yields that i.e. (2.13) holds.Lemma 2.3 (along with the previous remark) thus implies that the tilted measure µ h,V introduced in (1.8) is well-defined and satisfies the estimates (2.15) and (2.16) if λ 0 ă c 3 in (1.7).In fact, in the specific case of (2.17), the same conclusions could instead be derived by combining (1.19) with (2.18) below and the heat kernel bound (2.5).
Proof of Lemma 2.3.In view of (1.12), (1.14) and (1.15) and (1.7), observe that (with L " L 0,0 and notation we explain below) (2.18) ´L ě ´Lapϕq 2 ě ´c1 ∆ as symmetric positive-definite operators (restricted to Domp´∆q, tacitly viewed as a subset of R Z d ˆE independent of ϕ P E).Here, "A ě B" in (2.18) means that xf, Af y ě xf, Bf y for f P Domp´∆q where x¨, ¨y is the usual ℓ 2 pZ d q inner product; moreover ∆f pxq " ř y"x pf pyq ´f pxqq, for suitable f : Z d Ñ R (e.g.having finite support), so that p´∆q ´11 y pxq " 1 2d gpx, yq for all x, y P Z d with g " g 0 , cf. ( 2 , in the sense that the inequality holds for the restriction of either side to ℓ 2 pΛq with a constant c 1 uniform in Λ.This implies that the measure ν ξ Λ " µ ξ Λ,Q λ defined as in (1.3) but with H λ in place of H is log-concave and it yields, together with the Brascamp-Lieb inequality, uniformly in Λ and ξ, for suitable F (say depending on finitely many coordinates), where x¨, ¨y denotes the ℓ 2 pΛq inner product.In particular, choosing F " ϕ 0 and using that 2dp´∆q ´11 y pxq Õ gpx, yq ă 8 as Λ Õ Z d , one readily deduces from the resulting uniform bound in (2.20) and the Gibbs property (1.5) that e Q λ P L 1 pµq, and (2.15) then follows upon letting Λ Õ Z d in (2.20).
To obtain (2.16), one starts with (2.15) and introduces p´∆q ´1{2 (defined e.g. by spectral calculus) to rewrite the right-hand side of (2.15) up to an inconsequential constant factor as Writing the second moment on the right-hand side as a variance plus the square of its first moment and applying (2.15) once again to bound var µ Q λ ppp´∆q ´1{2 B ¨F qpxqq, (2.16) follows.
By iterating (2.15), one also has controls on higher moments.In view of Remark 2.4 above, the following applies in particular to µ h,V for any h, V as in (1.7).

Lemma 2.5. Under the assumptions of Lemma 2.3, for any
Defining cpkq " 0 for odd k and observing that M pkq vanishes for such k, (2.21) readily follows from (2.22) and a straightforward induction argument, with cp2kq " cpkq 2 `ck 2 cp2pk ´1qq.
3 Elements of potential theory for X ¨and intensity measure For the remainder of this article, we always tacitly assume that conditions (1.2) and (1.7) are satisfied for the data pU, h, V q.In this section, we develop various tools around the process X ẅith generator L given by (1.15).Among other things, these will allow us to define a natural intensity measure ν h,V on bi-infinite Z d ˆE-valued trajectories, see Theorem 3.5 below.This measure is fundamental to the isomorphism theorem derived in the next section.
We start by developing useful formulas for the equilibrium measure and capacity of "cylindrical" sets.For K a finite subset of Z d , abbreviated K ĂĂ Z d , we write Q K " K ˆE with E " R Z d for the corresponding cylinder and abbreviate is the discrete box of radius N .We use BK to denote the inner boundary of K in Z d and K c " Z d zK.Recalling Ep¨, ¨q from (1.16) with domain DpEq, we then define the capacity of Q K , for arbitrary K ĂĂ Z d , as ), along with various potentialtheoretic notions developed in the present section (e.g. e Q K , h Q k below), all implicitly depend on the tilt ph, V q.In view of (1.16), restricting to the class of functions f px, ϕq " f pxq satisfying the conditions in (3.1) but independent of ϕ, and observing that E µ h,V rapx, y, ϕqs ď c 2 for |x´y| " 1 due to (1.12) and (1.2), it follows that where cap Z d pKq refers to the usual capacity of the simple random walk on Z d .Similarly, neglecting the contribution from E 1 and applying Fatou's lemma, one obtains that We now derive a more explicit (probabilistic) representation of cappQ K q.Recalling that X t " pX t , ϕ t q stands for the process associated to E (with generator L " L h,V given by (1.15) and canonical law P px,ϕq , see below (1.17)), we introduce the stopping times H Q K " inftt ě 0 : and introduce, for suitable f :

)). The variational problem (3.1) has a unique minimizer given by
Proof.The property (3.7) follows by L-harmonicity of h Q K in view of (3.6).To see (3.8), denoting by pP t q tě0 the semigroup associated to X, one has for all z " px, ϕq P Q K , applying the Markov property at time t, which is plainly non-negative; to see that the limit on the right-hand side exists, denoting by τ the first jump time of X ¨, the spatial part of X ¨, one notes that it equals because X can only escape Q K through its spatial part X and the contribution stemming from two or more spatial jumps up to time t is Opt 2 q as t Ó 0; similarly the expectation in (3.10) is bounded by ct for t ď 1.
To obtain that h Q K is a minimizer, first note that by definition, see (3.4), and by transience, h Q K satisfies the constraints in (3.1).For arbitrary f as in (3.1), one has The first term in (3.11) is non-negative.On account of (1.16) and due to (3.6), (3.12) where the last step uses that h Q K p¨, ϕq " 1 on K, which is the support of e Q K p¨, ϕq, see (3.6).
The last expression in (3.12) is exactly the right-hand side of (3.9).To conclude, one observes that the third term in (3.11) can be recast using Epf ´hQ K , h q and the latter is non-negative because pf ´hQ K qp¨, ϕq ě 0 on K by (3.1).
A key ingredient for the construction of the intensity measure ν below is the following result.We write W QK below for the subset of trajectories in W `with starting point in Q K .Recall the definition of P ρ from (3.9) and abbreviate ρ " ρ h,V for the remainder of this section.

Proposition 3.2 (Sweeping identity).
With e Q K as defined in (3.6), for all K Ă K 1 ĂĂ Z d and bounded measurable f : To prove Proposition 3.2, we will use the following result.We tacitly identify E with the weighted L 2 -space E r " tϕ P E : |ϕ| 2 r def.

" ř
x |ϕ x | 2 e ´r|x| ă 8u for arbitrary (fixed) r ą 0, which has full measure under µ, and continuity on E is meant with respect to | ¨|2 r in the sequel.

Lemma 3.3 (Switching identity).
For all K ĂĂ Z d and v, w P C b c pZ d ˆEq (continuous bounded with compact support), where the last step uses that X ¨and X s´¨h ave the same law under P ρ .The last integral is readily seen to equal the expectation in second line of (3.13).
Proof of Proposition 3.2.For a given f as appearing in (3.13), consider the function v defined such that, with U as in (3.5), (i.e.let v " ´Lξ).By (3.9) and the strong Markov property at time Q K , one can rewrite In view of (3.15), (3.16), applying (3.14) with w " e Q K 1 and v as in (3.15) yields that the left-hand side of (3.13) equals Since K Ď K 1 , (3.6) and (3.4) imply that, on the event tH which yields (3.13).
Remark 3.4.The sweeping identity (3.13) corresponds to the classical Dobrushin-Lanford-Ruelle-equations in equilibrium statistical mechanics, see e.g.[37, Def.p.28]: for all K Ă K 1 Ă Z d and f " 1 tX 0 "zu , z P K, explicating (3.13) gives We now introduce the intensity measure ν which will govern the relevant Poisson processes.We write W for the space of bi-infinite right-continuous trajectories on Z d ˆE whose projection on Z d escapes all finite sets in finite time.Its canonical coordinates will be denoted by X t " pX t , ϕ t q, t P R, and we will abbreviate X ˘" pX ˘tq tą0 .We let W ˚" W { " be the corresponding space modulo time-shift, i.e. w " w 1 if pθ t wq " w 1 for some t P R, and denote by π ˚: W Ñ W the associated projection.We also write W Q K Ă W for the set of trajectories entering Q K , i.e. w P W Q K if X t pwq P Q K for some t P R, and W QK " π ˚pW Q K q.All above spaces of trajectories are endowed with their corresponding canonical σ-algebra, denoted by W, W ˚, W Q K etc.We then first introduce a measure ν Q K on pW , Wq as follows: with e Q K as defined in (3.6), and where, with a slight abuse of notation, we identify A ˘P σpX ˘q (part of W) with the corresponding events in W `. The latter is the σ-algebra of W `, the space of one-sided trajectories on which P px,ϕq is naturally defined.Note that the ρ-integral in (3.20) is effectively over A X tX 0 P Ku, hence ν Q K is a finite measure, and by (3.9), The family of measures tν Q K : K ĂĂ Z d u can be patched up as follows.
Theorem 3.5 (d ě 3, h, V as in (1.7)).There exists a unique σ-finite measure ν " ν h,V on pW ˚, W ˚q such that Proof.The uniqueness of ν follows immediately from (3.20), since for all A ˚P W ˚, with A " pπ ˚q´1 pAq, one has νpA ˚q " lim n ν Q Bn pA X W Q Bn q by monotone convergence.In order to prove existence, it is enough to argue that for all K Ă K 1 ĂĂ Z d and A ˚P W QK :

20), and, recalling
Hence, in order to obtain (3.23) it is sufficient to show that for all measurable A 0 P 2 K ˆE (where 2 K denotes the set of subsets of K) and To see why this implies (3.23), simply note that (3.24) corresponds to the choice A ˚" π ˚ptX 0 P A 0 , X `P A `uq in (3.23) with A 0 , A `as above, which generate W QK .It now remains to argue that (3.24) holds.By (3.20), the left-hand side of (3.24) can be recast as whereas the right-hand side of (3.24) equals of the measure ν h,V constructed in Theorem 3.5 is independent of V and h; indeed, in view of (1.12) and (1.14) the generator of the spatial component of P px,ϕq is that of a simple random walk.
The measure ν G obtained in this way is precisely (up to defining trajectories in continuous-time) the intensity measure of random interlacements constructed in Theorem 1.1 of [70].

An isomorphism theorem
We now derive a "Ray-Knight" identity for convex gradient Gibbs measures, which is given in Theorem 4.3 below.Recall that the measure ν " ν h,V defined by Theorem 3.5 depends implicitly on the choice of ph, V q appearing in (1.6), corresponding to the Gibbs measure µ h,V in (1.8).
In what follows, V will represent a (finite) region on which we seek to probe the field ϕ 2 sampled under µ " µ h"0,V "0 , cf. (1.4), corresponding to the observable xV, ϕ 2 y ℓ 2 pZ d q , and h will be carefully tuned with V in the relevant intensity measure, cf.(4.1) and (4.10).We now introduce these measures.Recall that we assume ph, V q to satisfy (1.7).For such h, V and all u ą 0, define the measure ν h,V u on pW ˚, W ˚q by (4.1) ν h,V u pAq " right-hand side of (4.1) can also be recast as 2u ´σqν σh,V pAq dσ).On account of Theorem 3.5, ν h,V u given by (4.1) defines a σ-finite measure.We can thus construct a Poisson point process ω on W ˚having ν h,V u as intensity measure.We denote its canonical law by P h,V u , a probability measure on the space of point measures Ω W ˚" tω " ř iě0 δ w i : w i P W ˚, i ě 0, and ω ˚pW QK q ă 8 for all K ĂĂ Z d u, endowed with its canonical σ-algebra F W ˚. The law P h,V u on pΩ W ˚, F W ˚q is completely characterized by the fact that for any non-negative, Of particular interest below is the corresponding field of (spatial) occupation times pL x q xPZ d , defined as follows: for ω " ř iě0 δ w i , let where w i P W is any trajectory such that π ˚pw i q " w i and X t pw i q is the projection onto the spatial coordinate of w i at time t.In what follows, we frequently identify V px, ϕq " V pxq, ϕ P E, viewed either as such or tacitly as multiplication operator pV f qpx, ϕq " V px, ϕqf px, ϕq, for suitable f .We first develop a representation of Laplace functionals for the field L that will prove useful in the sequel.
(Here, with hopefully obvious notation, 1 refers to the function of px, ϕq P Z d ˆE which is identically one).
Before going any further, let us first relate the above setup and the formula (4.4) to the (simpler) Gaussian case.when ω " ř iě0 δ w i .Classically, η is a Poisson process with intensity measure Π ˝νh,V u , and L x pωq " L x pηq, as can be plainly seen from (4.3).In the Gaussian case, substituting (3.27) into (4.1) and performing the integrals over τ and σ (note to this effect that ş ?2u 0 ş τ 0 dσdτ " u), one readily infers that η has intensity uν G , i.e. the law P G u of η is that of the interlacement process at level u ą 0, cf.[70].The field L x " L x pηq is then simply the associated field of occupation times (at level u).In this case, the formula (4.4) simplifies because the test function V is spatial and the dynamics generated by L 1 and L 2 decouple, see (1.10), (1.14) and (1.15).All in all Lemma 4.1 thus yields, for all V satisfying (1.7) and u ą 0, (4.5) where G V p" p´L a"1

2
´V q ´1q refers to the convolution operator on ℓ 2 pZ d q with kernel g V given by (2.2).On the other hand, one knows, see e.g.(2.11) in [71] in case V ď 0, that the left-hand side of (4.5) equals ´xV, pI ´GV q ´11y ℓ 2 , G " G V "0 , whenever }GV } 8 ă 1 (incidentally, note that (1.7) implies that }GV `}8 ă 1).With a similar calculation as that following (7.22)below, which is a continuous analogue, one can show that this expression equals the right-hand side of (4.5) when }GV } 8 ă 1.Notice however that (4.5) holds under the more general condition (1.7) as a result of Lemma 4.1, which places no constraint on V ´.
Proof of Lemma 4.1.The starting point is formula (4.2).First, note that by definition of the occupation times L ¨in (4.3), one can write xV, Ly ℓ 2 " ş W ˚fV ωpdw ˚q, where (recall that we tacitly identify V px, ϕq " V pxq, ϕ P E) Hence, applying (4.2) and then substituting (4.1), (3.22) and (3.20) for the intensity measure, one obtains, with K " supppV q, in view of (4.6), that strictly speaking, (4.2) does not immediately apply since V is signed but the necessary small argument using dominated convergence is readily supplied with the help of (2.4).It thus remains to be argued that the right-hand side of (4.7) equals that of (4.4).To this end, consider the function u t px, ϕq def.
which is bounded uniformly in t ě 0 on account of (2.4), and observe that, using first the fundamental theorem of calculus and then the Feynman-Kac formula for the process X, Dropping σh, V for ease of notation (i.e.writing L " L σh,V , ρ " ρ σh,V ), substituting (4.8) into (4.7) and noting that L `V is symmetric with respect to x¨, ¨yL 2 pρq , cf. (1.17), then yields that A e Q K p¨, ¨q, E p¨,¨q " e and (4.4) follows from (4.7) and (4.9) since V h Q K " V and xh Q K , V y L 2 pρq " x1, V y L 2 pρq on account of (3.4) (recall that K is the support of V ).
We now come to the main result of this section, which is the following theorem.Let (4.10) and write P V u " P h"V,V u for the canonical law of the associated Poisson point process on W ˚.
Recall that µ " µ h"0,V "0 refers to the Gibbs measure (1.4) for the Hamiltonian (1.1).With hopefully obvious notation, ϕ ¨`a for scalar a P R refers to the shifted field pϕ x `aq xPZ d below.

Theorem 4.3 (Isomorphism Theorem).
For all u ą 0 and V : We first make several comments.Remark 4.4.
1.One way to interpret Theorem 4.3 is as follows: the equality (4.11) holds trivially when u " 0. Thus, L ¨measures in a geometric way the effect of the shift ?2u on (squares of) the gradient field ϕ.
2. When U pηq " 1  2 η 2 (cf.(1.1)), Theorem 4.3 immediately implies the identity derived in Theorem 0.1 of [71], which is itself an infinite-volume analogue of the generalized second Ray-Knight identity given by Theorem 1.1 of [34].The relevant Poissonian law P V u " P u in the Gaussian case is the random interlacement point process introduced in [70].In the general (non-Gaussian) case, (4.11) does not give rise to an immediate identity in law due to the dependence of P V u on V .3. Our argument also yields a new proof in the Gaussian case U pηq " 1 2 η 2 .Indeed, whereas our proof proceeds directly in infinite volume, the proof of Theorem 0.1 in [71] exploits the generalized second Ray-Knight theorem, along with a certain finite-volume approximation scheme.Although we will not pursue this here, one could seek an argument along similar lines in the present context.In particular, this entails deriving a similar identity as (4.11) on a general finite undirected weighted graph with wired boundary conditions, thereby extending results of [34] (e.g. in the form presented in Theorem 2.17 of [72]) to the present framework.

It is of course tempting to investigate possible extensions of various others Gaussian isomor-
phism identities, see e.g. the monographs [54], [49], [72] for an overview, to convex gradient measures.We will return to the case of [48] and applications thereof elsewhere [25].
Proof.Expanding the square on the right-hand side of (4.11) and rearranging terms, we see that (4.11) follows at once if we can show that (4.12) where µ V " µ 0,V , cf. (1.8).The change of measure is well-defined given our assumptions (1.7) for V p¨q on account of Lemma 2.3.We rewrite the exponential functional appearing on the right-hand side of (4.12) as follows.Introducing the function where var µ τ V,V `xV, ϕy 2 ˘refers to the variance with respect to the tilted measure µ h,V , h " τ V .
Noting that f p0q " f 1 p0q " 0, expressing f p ? 2uq " f p ? 2uq ´f p0q in terms of its second derivative by interpolating linearly between τ " 0 and τ " ?2u and substituting (4.13), one obtains that Now, applying the Hellfer-Sjöstrand formula (1.19) to compute cov µ σV,V pϕ x , ϕ y q, recalling that V px, ϕq " V pxq, for all x P Z d , ϕ P E, and abbreviating L " L σV,V , it follows that 0 dt e tpL`V q V ˙p¨, ¨qF Putting together (4.15) and (4.14), one sees that the right-hand side of (4.12) is precisely the right-hand side of (4.4) for the choice h " V .Hence, the asserted equality in (4.12) follows directly from Lemma 4.1 on account of (4.10).

Renormalization and scaling limits of squares
We now aim to determine possible scaling limits for the various objects attached to Theorem 4.3, starting with linear and quadratic functionals of ϕ, as do appear e.g. when expanding the square on the right-hand side of (4.11).Our main result to this effect is Theorem 5.1 below, which will be proved over the course of the remaining sections.With ϕ the canonical field under µ, we introduce for integer N ě 1 the rescaled field (5.1) where tau " maxtk P Z : k ď au for a P R denotes integer part (applied coordinate-wise when the argument is in R d , as above) and for V P C 8 0 pR d q, set (5.2) Moreover, writing : X 2 :" X 2 ´Eµ rX 2 s for any X P L 2 pµq, let x: Φ 2 N :, V y def.
" : xΦ 2 N , V y : ´" ż R d V pzq : ϕ N pzq 2 : dz ¯. (5.3) (with : ϕ N pzq 2 :" ϕ N pzq 2 ´Eµ rϕ N pzq 2 s in the above notation).To avoid unnecessary clutter, we regard Φ k N , k " 1, 2 (as well as : Φ 2 N :q as distributions on R d , by which we always mean an element of pC 8 0 q 1 pR d q, the dual of C 8 0 pR d q, in the sequel.Indeed, xΦ k N , ¨y : C 8 0 pR d q Ñ R d is a continuous linear map; the topology on C 8 0 pR d q is for instance characterized as follows: f n Ñ 0 if and only if supppf n q Ă K for some compact set K Ă R d and f n and all its derivatives converge to 0 uniformly on K.We endow the space of distributions with the weak-˚topology, by which u n : C 8 0 pR d q Ñ R d converges to u : C 8 0 pR d q Ñ R d if and only if u n pf q Ñ upf q for all f P C 8 0 pR d q.Our main theorem addresses the (joint) limiting behavior of pΦ N , : Φ 2 N :q as N Ñ 8 when d " 3. Its statement requires a small amount of preparation.Recall that the Gibbs measure µ from (1.4) for the Hamiltonian (1.1) is translation invariant and ergodic.Hence, the environment a t p¨, ¨q " ap¨, ¨; ϕ t q in (1.12) generated by the ϕ-dynamics associated to µ (which solve (1.11) with V " h " 0) inherits these properties, and is uniformly elliptic on account of (1.2); that is, E µ P px,ϕq rc 1 ď a t p0, eq ď c 2 s " 1 for all t ě 0 and |e| " 1.By following the classical approach of Kipnis and Varadhan [44], see Proposition 4.1 in [38], one has the following homogenization result for the walk X ¨: with D " Dpr0, 8q, R d q denoting the Skorohod space (see e.g.[14,Chap. 3]), there exists a non-degenerate (deterministic) covariance matrix Σ P R dˆd such that, as n Ñ 8, the law of t Þ Ñ n ´1{2 X tn on D under E µ P px,ϕq p¨q tends to the law of a Brownian motion B " tB t : t ě 0u with B 0 " x, EpB t q " 0 and Eppv ¨Bt q 2 q " v ¨Σv, for v P R d . (5.4) The invariance principle (5.4) defines the matrix Σ.With G Σ p¨, ¨q denoting the Green's function of B, we further introduce the bilinear form (5.5) E Σ pV, W q " ż V pxqG Σ px, yqW pyq dx dy " xV, G Σ V y, for V, W P SpR d q, which is symmetric, positive definite and continuous (in the Fréchet topology).Hence, see for instance Theorem I.10, pp.21-22 in [66], there exists a unique measure P Σ on S 1 pR d q, characterized by the following fact: with Ψ denoting the canonical field (i.e. the identity map) on S 1 pR d q, under P Σ , for every V P SpR d q, the random variable xΨ, V y is a centered Gaussian variable with variance E Σ pV, V q. (5.6) We write E Σ r¨s for the expectation with respect to P Σ .The canonical field Ψ is the massless Euclidean Gaussian free field (with diffusivity Σ).
Of relevance for our purposes will be the second Wick power of Ψ.Let H be the (Gaussian) Hilbert space corresponding to Ψ, i.e. the L 2 pP Σ q-closure of txΨ, V y : V P SpR d qu.For X, Y P H, one defines the first and second Wick products as : X :" X ´EΣ rXs " X and : XY :" XY ´EΣ rXY s.For ρ ε,x p¨q " ε ´dρp ¨´x ε q, with ρ smooth, non-negative, compactly supported and such that ş ρpzqdz " 1, let Ψ ε pxq " xΨ, ρ ε,x y.The field : Ψ ε pxq 2 : is thus well-defined.Now let d " 3.For V P SpR 3 q, one can then define the L 2 pP Σ q-limits (5.7) x: Ψ 2 :, V y def.
" lim εÑ0 ż : Ψ ε pxq 2 : V pxq dx (elements of H) and one verifies that the limit in (5.7) does not depend on the choice of smoothing function ρ " ρ 1,0 .In what follows we often tacitly identify an element of S 1 pR 3 q with its restriction to pC 8 0 q 1 pR 3 q.The following set of conditions for the potential V will be relevant in the context of (5.3) and (5.7): (5.8) V P C 8 pR d q and for some λ ą 0, L ě 1, supppV q Ă B L and }V } 8 ď λL ´2.
For any value of λ ă c 5 (with a suitable choice of c 5 ą 0), one then obtains that r V t pz, z 1 q ď cr t pz, z 1 q for all t ě 0 and z, z 1 P R d and all V satisfying (5.8),where r t refers to the transition density of G Σ and r V t to that of its tilt by V (cf.(2.9) in case Σ " Id), which follows by a straightforward adaptation of the arguments in the proof of Lemma 2.1.In particular, this implies that for all W P C 8 0 pR d q, (5.9) (so G Σ " G 0 Σ , cf. above (5.5))whenever V satisfies (5.8), i.e.G V Σ acts (boundedly) on C 8 0 pR d q for such V , which is all we will need in the sequel.Associated to G V Σ in (5.9) is the energy form E V Σ p¨, ¨q defined similarly as in (5.5) with G V Σ p¨, ¨q in place of G Σ p¨, ¨q, whence E Σ p¨, ¨q " E 0 Σ p¨, ¨q.We now have the means to state our second main result, which identifies the scaling limit of Φ N , : Φ 2 N : introduced in (5.2)-(5.3).
Theorem 5.1 (Scaling limits, d " 3).As N Ñ 8, the law of pΦ N , : Φ 2 N :q under µ converges weakly to the law of pΨ, : Ψ 2 :q under P Σ .(5.10) Moreover, for all V, W P C 8 0 pR 3 q with V satisfying (5.8) with λ ă c, with E V Σ p¨, ¨q as defined below (5.9) and A V Σ pV, V q " ť V pzqA V Σ pz, z 1 qV pz 1 qdzdz 1 , where (5.12) The proof of Theorem 5.1 is given in Section 7 and combines several ingredients gathered in the next section.Remark 5.2.
1.The expressions on the right of (5.11) are well-defined, as follows from (5.9), the fact that G σV Σ pz, z 1 q ď cG Σ pz, z 1 q for all z, z 1 P R d and that G Σ pz, ¨q P L 2 loc pR 3 q, which together yield that A V Σ p¨, ¨q extends to a bilinear form on (say) C 8 0 pR 3 q (cf.Lemma 6.3 below).In particular, A V Σ pV, V q ă 8 for V as in (5.8) (and in fact sup V A V Σ pV, V q ď c). 2. Specializing to the case V " 0, Theorem 5.1 immediately yields the following 2 E Σ pW,W q , (5.13) (cf.(5.5) for notation), i.e.Φ N under µ converges in law to Ψ as N Ñ 8.
Corollary 5.3 is a celebrated result of Naddaf and Spencer, see Theorem A in [58], which has generated a lot of activity (see e.g.[16,21,2] for generalizations to certain non-convex potentials, [38] for extensions to the full dynamics tϕ t : t ě 0u, and [56] for a finitevolume version and [22,7] for quantitative results; see also [43] regarding similar findings for domino tilings in d " 2 and more recently [12,13] for the integer-valued free field in the rough phase; cf. also [32,33] and refs.therein for height functions associated to other combinatorial models.Thus, Theorem 5.1 extends the main result of [58] for d " 3. 3. Together, (5.10) and (5.11) imply in particular that for all V satisfying (5.8), (5.14) see also (7.24) below for a generalization of this formula to a non-zero scalar "tilt" u.
Explicit representations for moment-generating functionals of Gaussian squares usually involve (ratios) of determinants, see e.g.(5.46) in [54] or Proposition 2.14 in [72].We are not aware of any reference in the literature where (5.11) or (5.14) appear.
4. To illustrate the usefulness of these formulas, notice for instance that (5.11) immediately yields the following: The law of to a 'massive' free field with energy form (5.15) We refer to the proof of Corollary 7.5 below for another application of (5.11) in order to identify the scaling limit of the occupation-time field L appearing in Theorem 4.3.
5. Theorem 5.1 has no (obvious) extension when d ą 3. Indeed the existence of limits of renormalized squares as in (5.7) crucially exploits the local square integrability of the covariance kernel.
6.Although we won't pursue this here, by a slight extension of our arguments, one can state a convergence result akin to (5.10) but viewing pΦ N , : Φ 2 N :q as H ´spR 3 q ˆH´s 1 pR 3 q-valued, for arbitrary s ą 1 2 and s 1 ą 1.Here H ´spR 3 q denotes the dual of the Sobolev space H s pR 3 q " W s,2 pR 3 q endowed with the inner product pf, gq s " ş R 3 f pxqp1 ´∆q s gpxqdx, with ∆ denoting the usual Laplacian on R 3 .

Some preparation
In this section, we prepare the ground for the proof of Theorem 5.1.We derive three results, see Propositions 6.1, 6.7 and 6.9, organized in three separate subsections.Section 6.1, which contains Proposition 6.1, deals with exponential tightness of the relevant functionals (5.2) (when k " 1) and (5.3) (when k " 2).In Section 6.2 (cf.Proposition 6.7) we derive a key comparison estimate between quadratic functionals of Φ N and those of a certain smoothed field Φ ε N , to be introduced shortly, which is proved to constitute a good L 2 pµq-approximation of : Φ 2 N : for a suitable range of parameters.Finally, we show in Section 6.3 that the smoothed field behaves regularly, i.e. converges towards its expected limit (which actually holds for all d ě 3).Combining these ingredients, the proof of Theorem 5.1 is presented in the next section.
We now introduce the smooth approximation that will play a role in the sequel.Let ρ " ρ 1 be an arbitrary smooth, non-negative function with }ρ} L 1 pR d q " 1 having compact support contained in r´1, 1s d .For ε ą 0 and x P R d , let ρ ε p¨q " ε ´dρ 1 p ε q, ρ ε,z p¨q " ρ ε pz ´¨q.Define (6.1) " xΦ N , ρ ε,z y, z P R d and xpΦ ε N q k , V y, k " 1, 2, and x: pΦ ε N q 2 :, V y as in (5.2)-( 5.3) but with ϕ ε N in place of ϕ N .Note that z Þ Ñ ϕ ε N pzq inherits the smoothness property of ρ.The regularized field ϕ ε N essentially reflects at the discrete level the presence of an (ultraviolet) cut-off at scale ε in the limit.
6.1.Tightness.The main result of this section is Proposition 6.1, which implies in particular the exponential tightness of t: Φ 2 N :, N ě 1u, along with similar conclusions for its regularized version : pΦ ε N q 2 :, see (6.1) and Remark 6.2,(1)).The following bounds on Gaussian moments are interesting in their own right.We conclude this section by exhibiting how these estimates improve to exact calculations in the Gaussian case.For V, W P C 8 0 pR 3 q, let (6.2) Θ µ pχq def. " (whenever the integrand is in L 1 pµq) and recall ϕ N from (5.1) and that : X : " X ´Eµ rXs for X P L 2 pµq.The proofs of the following estimates will rely on Lemma 2.3.
Remark 6.2. 1.In particular, for any V, W as above, the random variables 1 2 x: 3) for notation, are (exponentially) tight by (6.3), and similarly for Φ ε N instead of Φ N using (6.4).Indeed, to deduce tightness observe for instance that by (6.3), E µ rcoshtx: Φ 2 N :, V y `xΦ N , W yus is bounded uniformly in N , from which the claim follows using the inequality e |x| ď coshpxq, valid for all x P R.
For instance, inspection of the proof below reveals that the constants can be chosen in a manner depending on }ρ} 8 only; see (6.15) below.
Proof.We first assume that W " 0 in (6.2) and will deal with the presence of a linear term separately at the end of the proof.Let ϕ 0 N " ϕ N , cf. (5.1) and (6.1), which will allow us to treat (6.3) and (6.4) simultaneously, the former corresponding to the case ε " 0 in what follows.The proof will make use of Lemma 2.3; we first explain how its hypotheses (2.12)-(2.14)fit the present setup.Consider the functional (6.5) which, up to renormalization, corresponds to the exponential tilt defining Θ µ pϕ ε N q in (6.2) (when W " 0).For ε " 0, recalling (5.1), one writes for all N ě 1, (6.6) with V N as in (2.6), which is of the form (2.12) with Q λ " diagpV N q.By assumption on V , cf. (5.8), supppV q Ă B L hence diampV N q ď N L.Moreover, that is, Q λ " diagpV N q satisfies (2.13) and (2.14) with R " N L, whenever λ ă c 3 , which we tacitly assume henceforth.The case ε ą 0 follows a similar pattern.Here one obtains using (6.1) that (6.5) has the form (2.12) and (2.13) is readily seen to hold with R " N pL `2q.All in all, it follows that e F ε N P L 1 pµq for all ε P r0, 1s on account of Lemma 2.3, which is in force.In particular, together with Jensen's inequality, this implies that Θ µ pϕ ε N q, ε P r0, 1s, as appearing on the left of (6.3)-(6.4), is well-defined and finite for all N ě 1.
For t P r0, 1s, define Θ µ pχ ; tq as in (6.2), but with ptV, 0q instead of pV, W q, whence Θ µ pχ ; 0q " 0 and Θ µ pχ ; 1q " Θ µ pχq.Observing that one finds, with a similar calculation as that leading to (4.14), where F ε N is given by (6.5) and (6.8) We now derive a uniform estimate (in N and t) for the variance appearing on the right-hand side of (6.7).We will use (2.16) for this purpose.For z, z 1 P R 3 and ε ě 0, let (6.9) ρ ε pz ´wq dw and define ρ 0 N pz, z 1 q " N 3 ¨1tNzu"tNz 1 u .Abbreviating B x " B Bϕx , one sees that for all x P Z d (6.10) B x ϕ ε N pzq " N 1{2 ¨N ´3ρ ε N pz, x{N q, z P R d (in particular, the right-hand side of (6.10) equals N 1{2 ¨1ttNzu"xu when ε " 0).Using (6.10), one further obtains that Now, one readily infers using (6.10) and the fact that ϕ is a centered field under µ ε t that E µ ε t rB x F ε N pϕqs " 0. Recalling the rescaled Green's function g N " g 0 N from (2.7), applying (2.16) with the choice µ " µ ε t and F " F ε N , observing that the first term on the right-hand side vanishes and substituting for B x B y F ε N , one deduces that where, for all ε ě 0, we have introduced (6.13) and we also used the fact that ρ ε N pz, z 1 q " ρ ε N pz, z 2 q whenever tN z 1 u " tN z 2 u, as apparent from (6.9).Note that (6.12) is perfectly valid for ε " 0, in which case g 0 N " g N as in (2.7) in view of (6.13) and the definition of ρ 0 N below (6.9).To complete the proof, it is thus enough to supply a suitable bound for the quantity in the last line of (6.12).To this effect, let pG ε N q k , k " 1, 2, (with pG ε N q 1 " G ε N ) denote the operator with kernel g ε N p¨, ¨qk , i.e. pG ε N q k f pzq " ş g ε N pz, z 1 q k f pz 1 qdz 1 , for any function f such that ş g ε N pz, z 1 q k |f pz 1 q|dz 1 ă 8 for all z P R d .The following result is key.Lemma 6.3.For all V P C 8 0 pR d q with supppV q Ă B L and ε P p0, 1q, sup and (6.14)-(6.15)hold for ε " 0 uniformly in N ě 1 with a constant c independent of ρ.
We postpone the proof of Lemma 6.3 for a few lines.Applying (6.15) to (6.12) and recalling the assumptions on V specified in (5.8), which are in force, it readily follows that var µ 0 t pF 0 N q ď cpLqλ 2 for all N ě 1, t P r0, 1s and var µ ε t pF ε N q ď cpL, ρqλ 2 for all N ě c 7 pεq, t P r0, 1s and ε P p0, 1s.Plugging these into (6.7), the asserted bounds (6.3) and (6.4) follow for W " 0.
The case W ‰ 0 is dealt with by considering με def." µ ε t"1 , the latter as in (6.8), and introducing for t P r0, 1s and ε P r0, 1s.Then, one defines Θµ pχ ; tq as in (6.2), but with pV, tW q instead of pV, W q and repeats the calculation starting above (6.7)with Θµ pχ ; tq in place of Θ µ pχ ; tq.The resulting variance of F ε N can be bounded using (2.15) (or (2.16) which boils down to the former since B x B y F ε N " 0) and (6.14).The bounds (6.3)-(6.4)then follow as Θµ p¨; t " 1q " Θ µ p¨q.
We now supply the missing proof of Lemma 6.3, which, albeit simple, plays a pivotal role (indeed, (6.15) is the sole place where the fact that d " 3 is being used).Before doing so, we collect an important basic property of the (smeared) kernel g ε N p¨, ¨q introduced in (6.13) that will be useful in various places.Recall that g ε N implicitly depends on the choice of cut-off function ρ " ρ 1 through ρ ε N , cf. (6.9).Lemma 6.4 (d ě 3).For all ε P p0, 1q and N ě ε ´1, (6.16) The proof of Lemma 6.4 is found in App.B. With Lemma 6.4 at hand, we give the Proof of Lemma 6.3.We show (6.15) first.By assumption on V , it is sufficient to argue that (6.17) sup uniformly in N ě cpεq (and for all N ě 1 with c 9 " 0 when ε " 0), from which (6.15) immediately follows.We first consider the case ε " 0, which is simpler.The fact that d " 3 now crucially enters.Recalling g N " g 0 N from (2.7), splitting the integral in (6.17) according to whether |z 1 | ď 1 N or not and arguing similarly as in the proof of Lemma 6.4 (see below (B.3)), one sees that for all z P R d and N ě 1, ż Bp0,Lq where volp¨q refers to the Lebesgue measure and the last bound follows as (6.18) This yields (6.17) for all N ě 1 when ε " 0. For ε ą 0 and all N ě ε ´1 one finds using (6.16) that ż and ż using (6.18) in the last step.Together, these bounds immediately yield (6.17).The proof of (6.14) follows by adapting the previous argument, yielding that ş Bp0,Lq g ε N pz, z 1 qdz 1 ď c}ρ} 2 8 L 2 uniformly in z P R d , L ě 1 and N ě cpεq, along with a similar bound when ε " 0. Remark 6.5.The case ε ą 0 in (6.5) could also be handled via a suitable random walk representation (with potential) when V ě 0. The latter is not a serious issue with regards to producing estimates like (6.3)-(6.4)since Θ µ can be bounded a-priori by replacing V by V `in (6.2).Now, letting Noting that Q ε N px, yq ě 0 when V ě 0, this leads to an effective random walk representation with finite-range (deterministic) conductances Q ε N px, yq which add to apϕq in (1.12).In particular, the lower ellipticity only improves.The potential V ε N is then seen to exhibit the correct scaling (e.g. it satisfies (2.3)).
We conclude this section by refining the above arguments in the Gaussian case.Indeed the proof of (6.3) (or (6.4)) can be strengthened in the quadratic case essentially because the variance appearing in (6.7) can be computed exactly.This improvement will later be used to yield the formula (5.11) in Theorem 5.1.
Thus consider a Gaussian measure µ G converging in law to ψ in the sense of (5.13).For concreteness, we define µ G to be the canonical law of the centered Gaussian field ϕ with covariance given by the Green's function of the time-changed process Y t " Z σ 2 t , t ě 0, where Z denotes the simple random walk, cf.above (2.1), and Σ " σ 2 Id with Σ the effective diffusivity from (5.4); see e.g.[56], Theorem 1.1 regarding the latter.Incidentally, σ 2 is proportional to E µ rU 2 pϕ 0 ´ϕe i qs for any 1 ď i ď d, which is independent of i by invariance of µ under lattice rotations.The following is the announced improvement over (6.3) for µ G .Proposition 6.6 (d " 3).For all V, W P C 8 0 pR 3 q with V satisfying (5.8) for λ ă c 10 , ( `AV Σ pV, V q `EV Σ pW, W q (see below (5.9) and (5.12) for notation).
Proof.Referring to µ G t as the measure in (6.8) with ε " 0 and µ " µ G , it follows using (6.7) that (6.20) Θ µ G pϕ N q " with F 0 N as defined in (6.5).We now compute the terms on the right-hand side of (6.20) separately.To avoid unnecessary clutter, we assume that σ 2 " 1.Using (5.1) and Wick's theorem, one finds that E µ G t rϕ ε N pzq 2 ϕ ε N pz 1 q 2 s " 2g tV N pz, z 1 q 2 `gtV N pz, zqg tV N pz 1 , z 1 q, where g tV N refers to the rescaled Green's function (2.7).Hence, Substituting these expressions into (6.20), the claim (6.19) follows by means of Lemma 2.2.
6.2.L 2 -comparison.With tightness at hand, the task of proving Theorem 5.1 requires identifying the limit.A key step is the following L 2 -comparison estimate, which implies in particular that : ϕ 2 N : and its regularized version : pϕ ε N q 2 : introduced in (6.1) are suitably close.More precisely, we have the following control.Recall that : X 2 : " X 2 ´Eµ rX 2 s for X P L 2 pµq.Proposition 6.7 (L 2 -estimate, ε P p0, 1q).For all V P C 8 0 pR 3 q such that supppV q Ă B L , there exists c 11 " c 11 pε, Lq P p1, 8q such that " 0, pd " 3q.
We start by collecting the following precise (i.e.pointwise) estimate for the kernel g ε N defined in (6.13), at macroscopic distances, which can be seen to play a somewhat similar role in the present context as Lemma 6.4 did to deduce tightness within the proof of Proposition 6.1.For purposes soon to become clear, we also consider (cf.(6.13)) (6.23) gε N pz, z 1 q " ż g N pz, wqρ ε N pw, z 1 q dw.
Indeed, for h " g 0 N or g ε N , this is (6.17), and the case h " gε N is dealt with similarly upon noticing that gε N pz, z 1 q ď c}ρ} 8 ε ´1 for |z ´z1 | ď 3ε.The latter is obtained in much the same way as the argument following (B.3): the absence of a mollification with ρ ε N from the left, cf.(6.23) and (6.13), will effectively make the first supremum on the right of (B.3) disappear; the rest of the argument is the same.Returning to (6.25), restricting to the set |z ´z1 | ď 3ε, bounding the kernel in (6.26) by a sum of positive kernels and applying (6.28) readily gives (6.28) sup (note that (6.28) is specific to d " 3; the rest of the proof isn't).We now consider the case |z ´z1 | ą 3ε, which exploits cancellations in (6.26).Adding and subtracting G (see above Lemma 6.8 for notation) in (6.26), using the elementary estimate a 2 ´b2 ď p|a| `|b|q|a ´b|, one sees that for all N ě 1 and ε ą 0, ĳ |V |pzqhpz, z 1 q|h 1 pz, z 1 q ´Gpz, z 1 q| |V |pz 1 qdzdz 1 (6.29) where h, h 1 P tg 0 N , g ε N , gε N u.Now, using (6.14) and its analogue for gε N , one obtains that (6.30) sup where, with hopefully obvious notation, Gε N is the operator with kernel gε N ; cf.above Lemma 6.3 for notation.Going back to (6.29), bounding |h 1 pz, z 1 q ´Gpz, z 1 q| by its supremum over |z ´z1 | ą 3ε and estimating the remaining integral over |V |pzqhpz, z 1 q|V |pz 1 q using (B.8) and (6.30), one sees that the right-hand side of (6.29) is bounded for N ě cε ´1 by |hpz, z 1 q ´Gpz, z 1 q|, where the sup is over h P tg 0 N , g ε N , gε N u, which in particular tends to 0 as N Ñ 8 on account of (6.24).Together with (6.25) and (6.28), this readily yields (6.21), for suitable choice of c 11 .
The proof of (6.22) is simpler.Proceeding as with (6.21), using (2.16) (or (2.15)), one obtains a bound of the form (6.25) where k ε N " g ε N ´g0 N .The proof then proceeds by adding and subtracting G, splitting the resulting integral and using (6.24) to control the long-distance behavior.
6.3.Convergence of smooth approximation.As a last ingredient for the proof of Theorem 5.1, we gather here the convergence of the smooth field ϕ ε N introduced in (6.1).This convergence is not specific to dimension d " 3.In a sense, (6.32) below can be viewed (at the level of finite-dimensional marginals) as a consequence of (5.13).Some care is needed to improve this convergence to a suitable functional level, which requires controlling the modulus of continuity of ϕ ε N .This will bring into play Lemma 2.5.Define the centered Gaussian field (6.31) Ψ ε pzq " xΨ, ρ ε,z y, z P R d where ρ " ρ 1 refers to the choice of mollifier above (6.1)and ψ is defined in (5.6).In the sequel we regard both the law of ϕ ε N " pϕ ε N pzqq zPR d under µ and Ψ ε " pΨ ε pzqq zPR d under P Σ as probability measures on C " CpR d , Rq (which is all the regularity we will need in the sequel), endowed with its canonical σ-algebra.Proposition 6.9 (d ě 3, ε P p0, 1q).

The law of ϕ ε
N under µ converges weakly to the law of Ψ ε under P Σ as N Ñ 8. (6.32) The proof of (6.32) will follow readily from the next two lemmas.We first establish convergence of finite-dimensional marginals and then deal with the regularity estimate needed to deduce convergence in C. Proof.For λ z P R, let W p¨q " ř zPK λ z ρ ε,z p¨q which is in C 8 0 pR d q by assumption on ρ, cf.above (6.1).Then by (5.13) λ z E Σ rΨ ε pzqΨ ε pz 1 qsλ 1 z using (6.31) and (5.6) for the last equality.Thus (6.33) holds.
To establish the required regularity, we use Lemma 2.5 to control higher moments.
Picking N ě c 1 pεq, one further ensures by means of (7.6) that the first term is, say, at most ε, yielding overall a bound on |E µ re :ξ N : s ´EΣ re :ξ: s| valid for all N ě c 1 pεq which is o ε p1q as ε Ó 0 on account of (7.3) and (7.9).Thus, (7.11) follows.
7.2.Scaling limit of occupation times and isomorphism theorem.We now return to Theorem 4.3, with the aim of identifying the limiting behavior of the identity (4.11).As a consequence of Theorem 5.1, we first deduce the existence of a limit for the occupation times L appearing in (4.11) under appropriate rescaling.With L " pL x q xPZ d as defined in (4.3) and for N ě 1, we consider (7.12) and the associated random distribution, with values in S 1 pR d q, defined by (7.13) xL N , V y " ż L N pzqV pzqdz, V P SpR d q.
We now introduce what will turn out to be the relevant continuous object.For u ą 0, we consider on a suitable space p p Ω, p F , r P Σ u q the S 1 pR d q-valued random variable r L, which is the occupation time measure at level u ą 0 of a Brownian interlacement with diffusivity matrix Σ.A formula for Laplace functionals of the random measure r L is given in Prop.2.6 of [73].We derive here a somewhat different identity which is more suitable to our purposes, cf. in particular (7.21) below.Recall that p´∆ Σ ´V q is invertible whenever V satisfies (5.8) with λ ă c 5 .
Proof.Applying the analogue of (4.2) for the Poisson measure r P Σ u , one finds using (7.14) and (7.15) along with the explicit formula for the intensity measure ν in [73, (2.3) and (2.7)], that where K Ą supppV q is a closed ball of suitable radius (recall supppV q is compact), E Σ x denotes expectation for Brownian motion on R d with diffusivity Σ (cf.(5.4)) started at x P R d , e K p¨q denotes its equilibrium measure on K (see e.g. in [69,Prop. 3.3] for its definition in the present context) and G V Σ " p´∆ Σ ´V q ´1, cf.(5.9).As G Σ e K " 1 on K " supppV q where G Σ " G 0 Σ , one has xV, 1y " xe K , G Σ V y and (7.16) follows upon noticing that (omitting superscripts Σ) 2).The following relates the fields L N and r L in (7.12) and (7.15).
2. Let L 0 N denote the occupation time measure defined as in (7.12)-(7.13)but for the interlacement process with intensity measure uν V "0,h"0 .Let P 0 u denote its law.Then one can in fact show that for all d ě 3, with u N as in Theorem (7.5), (7.25) L 0 N under P 0 u N converges to r L under r P Σ u as N Ñ 8 (as random measures on R d ).The limit (7.25) can be obtained by starting from the analogue of (7.16) for L 0 N by exploiting the invariance principle 5.4 directly and e.g. the bounds of [23] to deduce convergence to the right-hand side of (7.16).We omit the details.
3. It is instructive to note that the proof of Theorem 5.1 only relied on two 'external' ingredients, Lemma 2.3 (a consequence of (2.18)) and Theorem 5.3.Whereas the lower ellipticity seems difficult to get by, the upper ellipticity assumption in (1.7) can be reduced.For instance, using the results of [4,6], it follows that Theorem 5.1 continues to hold if only c ď V 2 and E µ rV 2 pBϕpeqq p s ă 8, for all edges e P te i , 1 ď i ď du and large enough p ą 1.
4. It would be interesting to obtain an analogue of Theorem 5.1 in finite volume, much in spirit like the extension by Miller [56] of the result of Naddaf-Spencer [58], cf.Theorem 5.3.
It would be equally valuable to seek such results for potentials with lower ellipticity, such as those appearing in [15] and [57].Suitable extensions of Brascamp-Lieb type concentration inequalities, such as those recently derived in [53], may plausibly allow to extend the tightness and L 2 -estimates in Propositions 6.1 and 6.7 to setups without uniform convexity.
A Heat kernel bounds with potential and scaling limits We collect here the proofs of Lemmas 2.1 and 2.2, which concern estimates for the tilted kernel q V t and the corresponding Green's function g V introduced in (2.1) and (2.2), along with scaling limits of the latter.
Proof of Lemma 2.1.By monotonicity, it is enough to consider the case V " V `, which will be assumed from here on.We first explain how (2.4) implies (2.5).For all t ě 0 and x, y, P Z d , using the inequality ab ď 1 2 pa 2 `b2 q, applying time-reversal and the Markov property, one obtains By a standard on-diagonal estimate, it follows from (A.1) that (A.2) .4) in the last step.To deduce (2.5), one applies the Cauchy-Schwarz inequality and a well-known lower bound on q t to deduce, for all t ě 0 and x, y P Z d , q V t px, yq ď q 2V t px, yq 1{2 q t px, yq 1{2 (A.2) ď cpt _ 1q ´d{2 q t{2 px, yq 1{2 ď c 1 q t{2 px, yq.
We now show (2.4).Let r " diampsupppV q.By translation invariance, we may assume that supppV q Ă B r " pr´r, rs X Zq d .Assume that (2.3) for some ε ą 0 to be determined, which translates to V ď ε r 2 .Then, with T B " inftt ě 0 : Z t R Bu denoting the exit time from B Ă Z d , for all x P Z d , one obtains whenever ε ď c α 2 for some small enough c P p0, 1q, using that sup x,N ě1 E x re cN ´2T B N s ď c 12 in the last step.Now consider the sequence of successive return times to B r and departure times from B αr : i.e., R 1 " H Br " inftt ě 0 : Z t P B r u and for each k ě 1, define D k " T Bαr ˝θR k `Rk (with the convention that D k " 8 whenever R k " 8) and R k`1 " R 1 ˝θD k `Dk (with a similar convention), where θ s , s ě 0, denote the canonical shifts for Z.Moreover, let (A.4) γpαq def.
Next, we prove Lemma 2.2, which is employed within the proof of Proposition 6.6 for the computation of the limiting generating functionals in the Gaussian case.
Proof of Lemma 2.2.Let L 1 be such that supppf q Ă r´L 1 , L 1 s d .Combining Lemma 6.3 in case ε " 0 with the bound (2.5) (note to this effect that the condition (2.3) applies with the choice V " V N uniformly in N ě 1 whenever V satisfies (5.8)), it follows that › › G V N f › › 8 ď cpL, L 1 q}f } 8 uniformly in N for all d ě 3, along with a similar bound for pG V N q 2 when d " 3. The same conclusions apply to G V , pG V q 2 .We now show (2.10).Recalling (2.7), rescaling time by N ´2 and using translation invariance of P x , one rewrites for arbitrary T ą 0 and all N ě 1, with Z N t " 1 N Z N 2 t the diffusively rescaled simple random walk (cf.above (2.1) for notation), (A.5) xf, G V N f y " a N pT q `bN pT q, where a N pT q " N ´d ż r0,1q d ˆr0,1q d ı and b N pT q is the corresponding expression with integral over t ranging from rT, 8q instead.Note that by assumption on f , the sum over x is effectively finite and restricted to x satisfying |x| 8 ď N L 1 .Using the fact that the functions f p x N `z N `¨q for |x| 8 ď N L 1 , z P r0, 1q d , are equicontinuous and uniformly bounded and applying the invariance principle for Z together with a straightforward Riemann sum argument, one concludes that for all T ą 0, (A.6) a N pT q To deal with b N pT q one applies the heat kernel estimate (2.5) (to V " V N ), thus effectively removing the tilt e As the right-hand side of (A.7) tends to 0 as T Ñ 8, (A.6) and (A.7) yield (2.10).
The last integral is bounded by considering the cases |v ´w| ď 1 N and 1 N ď |v ´w| ď c 13 ε separately (note that this is well-defined as 1 N ď ε) and bounding g N p¨, ¨q ď cN in the former case while using that g N pv, wq ď Upon being multiplied by ε ´d and uniformly in N ě ε ´1 the first of these terms is of order ε ´1 while the second one is of order ε 2´d , which is larger as d ě 3. Feeding the resulting bound into (B.3) and using (B.2) is then seen to imply that g ε N pz, z 1 q ď c 1 }ρ} 2 8 ε 2´d for N ě cε ´1, as desired.
We continue with the Proof of Lemma 6.8.We consider the case h ε N " g ε N first and discuss how to adapt the following arguments to the case of gε N at the end of the proof.Let G ε " ρ ε ˚G ˚ρε where ˚denotes convolution on R d , i.e. pf ˚gqpxq " ş f px ´yqgpyqdy for suitable f, g (note that G ε is welldefined since G acts as a convolution operator on C 8 0 pR d q Q ρ ε ˚ρε ).The function y Þ Ñ Gpx, yq being harmonic for all y P R d ztxu, one readily deduces using the mean-value property and the fact that ρ ε p¨q is supported on Bp0, εq that (B.4) G ε py, zq " Gpy, zq for all |y ´z| ą 2ε.
Next, observe that (B.7) ˇˇpg ε N q 1 py, zq ´pg ε N q 2 py, zq ˇˇď ż ´ż ρ ε py ´y1 qg N py 1 , z 1 qdy 1 ¯ˇρ ε N pz, z 1 q ´ρε pz ´z1 q ˇˇdz 1 By (6.9) the integrand in (B.7) (as a function of z 1 alone) tends to 0 pointwise as N Ñ 8 for all z 1 .Moreover for any f P C 8 0 pR d q with supppf q Ă Bp0, Rq, denoting by G N the convolution operator with kernel g N , one has that Going back to (B.7) and using (B.8), letting R " diampsupppρ ε qq, the integrand on the righthand side is thus bounded uniformly in N (and z) by ρ ε pz ´vq P L 1 pdz 1 q and it follows by dominated convergence that (B.9) lim N sup |y´z|ą3ε ˇˇpg ε N q 1 py, zq ´hε N py, zq ˇˇ" 0 for h ε N " pg ε N q 2 .The conclusion (B.9) continues to hold if one chooses h ε N " g ε N instead, for then ρ ε py ´y1 q on the right-hand side of (B.7) must be replaced by ρ ε N py, y 1 q and the rest of the argument still applies since sup N ě1,yPR d }ρ ε N py, ¨q} 8 ă 8, as required to obtain a uniform upper bound in (B.8).Together, (B.9), (B.6) and (B.4) yield (6.24) for h ε N " g ε N .To deal with h ε N " gε N , one considers Gε " G ˚ρε instead of G ε (in particular (B.4) continues to hold) and introduces pg ε N q 2 as in (6.23) but with ρ ε in place of sole occurrence of) ρ ε N .One then separately bounds |pg ε N q 2 ´G ε | and |pg ε N q 2 ´g ε N | much as in (B.5) and (B.7), respectively, but the details are simpler due to the absence of the integral over dy 1 .This completes the proof.

C Some Gaussian results
In this section we prove Lemma 7.3, which is a purely Gaussian claim used in the course of proving Theorem 5.1.We start with a preparatory result.For δ ą 0 and z P R 3 , define z δ to be the unique element x P δZ 3 such that z P x `r0, 1 δ q 3 .Recalling Ψ ε from (6.31), let Ψ ε δ be the Gaussian field defined by Ψ ε δ pzq " Ψ ε pz δ q, z P R 3 .The following is tailored to our purposes.
We conclude with the

26 )
But by Proposition 3.2, the right-hand side of (3.25) and (3.26) coincide, and (3.24) follows, which completes the proof.Remark 3.6.Let Π : W ˚Ñ W ˚denote the projection onto the first (Z d -valued) component of a trajectory, i.e.W is the space of bi-infinite Z d -valued transient trajectories.In the Gaussian case U pηq " 1 2 η 2 , cf. (1.1), the projection

Remark 4 . 2 .δ
With Π denoting the projection onto the spatial component (cf.Remark 3.6 for its definition), consider the induced process η " Πpωq def.Πpw i q
Hence, applying the strong Markov property successively at times R k and D k´1 , it follows that