The scaling limit of the KPZ equation in space dimension 3 and higher
Jacques Magnen and Jérémie Unterberger
Centre de Physique Théorique,111Laboratoire associé au CNRS UMR 7644 Ecole Polytechnique,
91128 Palaiseau Cedex, France
Institut Elie Cartan,222Laboratoire associé au CNRS UMR 7502. J. Unterberger acknowledges the support of the ANR, via the ANR project ANR-16-CE40-0020-01. Université de Lorraine,
B.P. 239, F – 54506 Vandœuvre-lès-Nancy Cedex, France
We study in the present article the Kardar-Parisi-Zhang (KPZ) equation
in dimensions in the perturbative regime, i.e. for small enough and a smooth, bounded, integrable initial condition . The forcing term in the right-hand side is a regularized space-time white noise. The exponential of – its so-called Cole-Hopf transform – is known to satisfy a linear PDE with multiplicative noise. We prove a large-scale diffusive limit for the solution, in particular a time-integrated heat-kernel behavior for the covariance in a parabolic scaling.
The proof is based on a rigorous implementation of K. Wilson’s renormalization group scheme. A double cluster/momentum-decoupling expansion allows for perturbative estimates of the bare resolvent of the Cole-Hopf linear PDE in the small-field region where the noise is not too large, following the broad lines of Iagolnitzer-Magnen . Standard large deviation estimates for make it possible to extend the above estimates to the large-field region. Finally, we show, by resumming all the by-products of the expansion, that the solution may be written in the large-scale limit (after a suitable Galilei transformation) as a small perturbation of the solution of the underlying linear Edwards-Wilkinson model () with renormalized coefficients .
Keywords: KPZ equation, Cole-Hopf transformation, directed polymer, constructive field theory, renormalization, cluster expansion, resolvent, large deviation estimates.
Mathematics Subject Classification (2010): 35B50, 35B51, 35D40, 35K55, 35R60, 35Q82, 60H15, 81T08, 81T16, 81T18, 82C41.
- 0 Introduction
- 1 Cole-Hopf transform
- 2 Multi-scale expansion and vertex representation
- 3 Cluster expansions
- 4 Renormalization
- 5 Final bounds
- 6 Appendix. Cluster expansions
- 7 Appendix. The effective propagator
The KPZ equation  is a stochastic partial differential equation describing the growth by normal deposition of an interface in space dimensions, see e.g. [7, 19]. By definition the time evolution of the height , , is given by
where is a regularized white noise, and are constant. Three terms contribute to eq. (0.1): a viscous term proportional to the viscosity , leading to a smoothening of the interface; a growth by normal deposition with rate , called deposition rate, and playing the rôle of a coupling constant; and a random rise or lowering of the interface modelling molecular diffusivity, with coefficient called noise strength. In a related context, also represents the free energy of directed polymers in a random environment [43, 20, 26]. It makes sense to consider more general nonlinearities of the form with , say, positive and convex, instead of , which is in any case an approximation of , assuming that the gradient (the slope of the interface) remains throughout small enough so that the evolution makes physically sense, precluding e.g overhangs.
The interest is here in the large-scale limit of this equation, for and/or large. A well-known naive rescaling argument gives some ideas about the dependence on the dimension of this limit. Namely, the linearized equation, a stochastic heat (or infinite-dimensional Ornstein-Uhlenbeck ) equation called Edwards-Wilkinson model  in the physics literature,
– where requires no regularization – is invariant under the rescaling ; we used here the equality in distribution, . Assuming that is a solution of the KPZ equation instead yields after rescaling
where (up to change of regularization) . For , vanishes in the limit ; in other terms, the KPZ equation is infra-red super-renormalizable, hence (power-like) asymptotically free at large scales in dimensions, i.e. expected to behave, in a small coupling (also called small disorder) regime where , like the corresponding linearized equation up to a redefinition (called renormalization) of the diffusion constant and of the noise strength .
Let us emphasize the striking difference with the one-dimensional equation. For this equation, scaling behaviors, see (0.3), are reversed with respect to , in other words, KPZ is (power-like) asymptotically free at small scales (i.e. in the ultra-violet), or equivalently (in the PDE analysts’ terminology) sub-critical. A large part of the interest for this equation comes from the fact that the large-scale strongly coupled theory [3, 26] is understood by comparison with integrable discrete statistical physics models [27, 64, 68, 71] relating to weakly asymmetric exclusion process  or the Tracy-Widom distribution of the largest eigenvalue of random matrices connected with Bethe Ansatz , free fermions and determinantal processes ,… Note that is believed by perturbative QFT arguments to be strongly coupled at large scales [7, 19] and its large-scale limit is not at all understood.
We prove the diffusive limit of -dimensional KPZ with small coupling in the present work, thus establishing on firm mathematical ground old predictions of physicists, see e.g. Cardy . The space dimension does not really matter as long as . In the small-coupling regime, contrary to the -case, we fall into the Edwards-Wilkinson universality class.
In comparison with the achievements made in the study of strongly coupled large-scale , this problem looks at first sight of lesser importance and difficulty. We believe that the interest of our result lies in the precision of our asymptotics, and in the potential wide scope of applicability of our methods.
Namely, the KPZ model is one particular instance of a large variety of dynamical problems in statistical physics, modelized as interacting particle systems, or as parabolic SPDEs heuristically derived by some mesoscopic limit, which have been turned into a functional integral form analogous to the Gibbs measure of equilibrium statistical mechanics, , using the so-called response field (RF), or Martin-Siggia-Rose (MSR) formalism and studied by using standard perturbative expansions originated from quantum field theory (QFT); for reviews see e.g.  or . Despite the lack of mathematical rigor, this formalism yields a correct description of the qualitative behaviour of such dynamical problems in the large scale limit.
The Feynman perturbative approach, see e.g. , consists in expanding into a series in and making a clever resummation of some truncation of it into so-called counterterms, represented in terms of a sum of diagrams; as such, it is non-rigorous, since it yields -point functions in terms of an asymptotic expansion in the coupling parameter which is divergent in all interesting cases (at least for bosonic theories). A few years ago, however, Gubinelli, M. Hairer, H. Weber,… [38, 39, 40, 6, 17, 18, 22, 23, 41, 59, 21], drawing sometimes on a dynamical approach to the construction of equilibrium measures advocated by Nelson , Parisi-Wu , and Jona-Lasinio, Mitter and Sénéor [45, 46, 47], have started developing this philosophy in a systematic way to solve sub-critical parabolic SPDEs rigorously, i.e. beyond perturbation theory. Such SPDEs have only a finite number of counterterms, each counterterm being the sum of a finite number of terms (that can be interpreted in terms of Feynman diagrams), which makes the task considerably easier, but still far from trivial.
Constructive approaches developed in the context of statistical physics by mathematical physicists from the mid-60es, see e.g. [28, 32, 33, 30, 31, 36, 42, 56, 57, 60] and surveys [35, 58, 66, 67, 72], have developed sophisticated, systematic truncation methods making it possible to control the error terms. The partial resummations are interpreted in the manner of K. Wilson [75, 76] as a scale-by-scale, finite renormalization of the parameters of the Lagrangian . In many instances it has proved possible to subtract scale counterterms explicitly by hand and prove that the remainder is finite, yielding some description of the effective, large-scale theory, see e.g. works in diverse contexts – random walks in random environment, KAM theory, etc. – by Bricmont, Gawedzki, Kupiainen and coauthors [15, 14, 16], and recent extensions to the study of sub-critical parabolic PDEs [51, 52], as an alternative to the ”global counterterm” strategy mentioned in the last paragraph. However, the implementation of a full-fledged, multi-scale constructive scheme is for the moment limited to equilibrium statistical physics models.
The present work is, to the best of our knowledge, the first attempt to use such a scheme in the context of non-equilibrium statistical mechanics, here for a parabolic SPDE. Instead of using the MSR formalism, we develop (as all previously mentioned mathematically rigorous approaches do) a more straightforward approach, starting directly from the equation and cutting the propagator into scales. We actually work on the following model.
The model. Let . We consider the following equation on ,
where is a white noise regularized in time and in space; is a smooth, bounded, integrable initial condition, i.e. are ; is small enough; and is a constant, average interface velocity which we shall fix later on.
The precise choice of regularization for the white noise is unimportant; one should just keep in mind that local (in time and space) solvability of (0.1) in a strong sense requires that, for every compact set (equivalently, for any as in Definition 2.1 (iii)), is locally integrable. For simplicity of exposition, we define to be a smooth, stationary Gaussian noise with short-range covariance. To be definite:
we fix a smooth, isotropic (i.e. invariant under space rotations) function with support and -norm , and let
Our main result is the following. Gaussian expectation with respect to is denoted either by , or or also if one wants to emphasize the dependence on the parameters ; the result also depends obviously on the initial condition . By convention, refers to the expectation with respect to the measure of the Edwards-Wilkinson equation with zero initial condition, where is a standard (unregularized) space-time white noise; for this equation we implicitly set . By definition, is a centered Gaussian process.
Let . Fix and a smooth, bounded, integrable initial condition . Let be small enough, . Then there exist three coefficients , and , all independent of the initial condition , such that the solution of the KPZ equation (0.4) satisfies the following asymptotic properties:
for all with ,
for all , with , and , letting ,
Since is a Gaussian measure, 2. may be rephrased as follows. Let
(, ). Then
where the sum ranges over all pairings of the indices .
In other words, up to a Galilei transformation , the -point functions of the KPZ equation behave asymptotically in the large-scale limit as the -point functions of the solution of the Edwards-Wilkinson equation with renormalized coefficients ,
where requires no regularization. Generally speaking, main corrections to the above asymptotic behaviour (0.6,0.10) are smaller by as proved in §5.3 D. Effective coefficients have a (diverging) asymptotic expansion in terms of ; lowest-order corrections in are computed in (4.29) and (LABEL:eq:leading-D-eff). The -term in (0.6) is a contribution due to the initial condition; further contributions of the initial condition to -point functions come with an extra multiplicative factor in , which is the scaling of the vertex. Corrections to Gaussianity of -point functions, of order , are examined in (2) a few pages below. Furthermore, our multi-scale scheme actually involves an effective propagator differing slightly from the effective Edwards-Wilkinson propagator , see section 7; this implies a correction w.r. to the r.h.s. of (0.10) with a small extra prefactor, which is proved to be a but could easily be improved to with arbitrary large.
Remark. A more common choice of regularization for is to take a discretized ”kick force”, namely, we pave by unit size intervals , , and let , be independent, centered Gaussian fields on which are constant in time and have smooth, space-translation invariant covariance kernel with finite range, for instance. This does not change the conclusion of Theorem 0.1, except that, the law of being now only -periodic in time, is now a -periodic function instead of the constant . This regularization has several advantages (see section 1); it allows in particular an explicit representation of in probabilistic terms. The scheme of proof extends without any significant modification if the covariance kernel decreases heat-kernel-like in space, e.g. if where is a standard space white noise, and is some constant.
Furthermore, it follows from the proof (see section 5) that the value of may be obtained by equating it to the constant such that independently of , in coherence with the value obtained in Carmona-Hu  in a discrete setting for a random directed polymer measure (see section 2.1), where is the Cole-Hopf transform of (see below). Let us note that the equality between and points out to the fact that we are in a weak disorder regime in which the annealed and quenched free energies coincide. However, our proof is independent of that of Carmona and Hu (see , Theorem 1.5), based on Gaussian concentration inequalities.
The proof follows closely the article by Iagolnitzer-Magnen  on weakly self-avoiding polymers in four dimensions, which is the main reference for the present work. Namely, up to the change of function (called Cole-Hopf transform) and of coupling constant, , the KPZ equation is equivalent to the linear equation , solved as , where is a random resolvent. Formally then, our problem is a parabolic counterpart to the large-scale analysis of polymers in a weak random potential solved in  by studying the equilibrium resolvent , where the ””-coefficient is the Edwards model representation of the self-avoiding condition (the model is solved for but the self-avoiding condition is recovered for ). Though the two models are physically unrelated, one must analyze similar mathematical objects. As is often the case, the model with a time evolution (i.e. the parabolic one) turns out to be easier than the equilibrium model (i.e. the elliptic one), because of the causality constraint.
The general scheme of proof, following, as mentioned above, the philosophy of constructive field theory, is to introduce a multi-scale expansion and define a renormalization mapping, , or equivalently (later on interpreted as the flow of the coupling constant through the Cole-Hopf transform), ensuring the convergence of the expansion at each scale and allowing to control error terms. The average interface velocity is fixed by requiring that the asymptotic velocity vanishes. The original parameters , called bare parameters, describe the theory at scale , while the Edwards-Wilkinson model with scale parameters and drift velocity give a good approximation of the theory at time distances of order , which becomes asymptotically exact in the infra-red limit, when . This goal is achieved in general by using a phase-space expansion, i.e. a horizontal cluster expansion casting into the form of a series the interactions at a given energy-momentum level between the degrees of freedom, and a vertical cluster or momentum-decoupling expansion separating the different energy-momentum levels. Energy, resp. momentum, are the Fourier conjugate variables of time and space; here a given energy-momentum level is adequately defined by considering heat-kernel propagators
with . Then the above series (roughly speaking, a truncated power series in the coupling constants with a bounded integral, Taylor-like remainder) converge if the bare coupling constant is small enough.
With our choice of covariance function for , however, the flow of the parameters is actually trivial starting from , i.e. for , and the noise strength , defined by resumming connected diagrams with four external legs, though scale-dependent, requires no renormalization at all, because the equation is infra-red super-renormalizable, and the total correction (obtained by summing over scales) is finite. This, and also the causality condition preventing the so-called low-momentum field accumulation problem [36, 30, 72], leads to a much simplified framework, from which the phase space analysis has almost disappeared. Only scale 0, two-point diagrams need to be renormalized, with a contribution at near zero momentum
leaving a remainder of parabolic order three in the momenta, i.e. or . Scale 0 diagrams are connected by ”low-momentum” heat-kernel propagators with , . A crucial point in the proof is that, thanks to the , remainders integrated over space-time cost a factor , namely (see (2.19) and (5.21))
or, simply said, . What is left of the cluster expansions is adequately resummed as in  into the random resolvent in the form of localized ”vertex insertions” (see section 5), thereby suppressing combinatorial factors which make the series divergent. Then the contribution of all vertex insertions is bounded by some contour integral of a modified resolvent through the use of Cauchy’s formula.
An extra complication comes however from the inverse Cole-Hopf transform. Applying cluster expansions – which is done in practice by differentiation with respect to some additional parameters – to leads to rational expressions of the form , where the ’s are differential operators, acting on ”replicas” of . Then the scale 0 diagrams requiring renormalization can be factorized, hence averaged with respect to the measure . Remaining terms are shown to yield a convergent series in the form of a sum over ”polymers” for small enough.
The and -pre-factors contained in Theorem 0.1 may be guessed from the following guiding principles, put into light by the cluster expansion.
(1) First, the two-point function of the renormalized Edwards-Wilkinson equation,
(), scales like , as can be seen by simply rescaling variables in the integral. There are two regimes: the equilibrium regime (), in which is essentially the equilibrium Green function of the Laplacian; the dynamical regime (), in which .
(2) The connected quantities (also called truncated -point functions) are . Namely, Gaussian pairwise contractions yield the expected scaling in , i.e. per link, as expected from (1); whereas the connected expectation requires supplementary links and twice as much vertices (since these are not present in the linear theory) in the expansion, contributing an extra small prefactor. The cluster expansion makes it possible to develop those links explicitly.
The plan of the article is as follows. We start by recalling the Cole-Hopf transform in section 1, and make the bridge to previous results on the subject stated in terms of the associated directed polymer measure. We then introduce in section 2 a multi-scale expansion for the propagators, together with multi-scale estimates (also called ”power-counting”), which are the building blocks of our approach. Sections 3, 4, and 5 are the heart of the article. The dressed equation, and the cluster expansion thereof, is presented in section 3. Section 4 is dedicated to renormalization; the scale 0 counterterms obtained by factorizing two-point functions through a supplementary Mayer expansion are bounded. Then we show in section 5 how to bound the sum of all terms produced by the expansion, and obtain final bounds for -point functions, proving thus our main result, Theorem 0.1. Finally, there are two appendices. In the first one, we provide detailed combinatorial formulas for the horizontal and Mayer cluster expansions. The second one is merely dedicated to a technical result. Pictures are provided, which are there to help the reader visualize the outcome of the various expansions.
(parabolic distance) Let . Similarly, for , , (Hausdorff distance). Then is the space projection of the distance , i.e. , etc.
Let be two functions on some set . We write if there exists some inessential constant (possibly depending on the parameters and on the space dimension ), uniform in for small enough, such that . Then, by definition, . If and , we write .
In many situations, one obtains -dependent functions such that decays Gaussian-like, for some positive constant bounded away from . We then write without further specifying the value of , which may change from line to line. For instance, if is the heat kernel, then we may write , leaving out the dependence in the parameter as explained in 2. Note however that, if , , whereas the inequality does not hold uniformly in because the space decay of is slower than that of .
Acknowledgements. We wish to thank H. Spohn, F. Toninelli and the referee for numerous discussions, suggestions and corrections, which have hopefully contributed in particular to the readability of the paper.
1 Cole-Hopf transform
It is well-known that is a solution of the linear equation with multiplicative noise,
plays the rôle of a bare coupling constant, from which (representing the solution as a Wiener integral by Feynman-Kac’s formula)
where the expectation is relative to the Wiener measure on -dimensional Brownian paths issued from with -normalization, i.e. , . Thus may be interpreted as the partition function of a directed polymer, see e.g.  and references within, but we shall not need this interpretation in the article. Note that , where is now a standard Brownian motion, from which – forgetting about the regularization and using the variable instead of –
Thus may be expanded in a series in the parameter . Similarly, , or conversely , from which
Without using the general theory developed in [73, 74], eq. (1.3) and (LABEL:eq:4.3) show that a.s. , exist and are for , say, and compactly supported. The Cole-Hopf solution coincides with the solution defined for more general Hamilton-Jacobi equations in [73, 74].
For the rest of the subsection only, we assume that is a discretized ”kick force”, i.e. are independent and constant in time, in order to compare with the existing literature. Since are independent fields, letting , where
leads to for any and if , whence more generally
Expanding the exponential in (1.5) and using
one gets: , whence .
Let us state an easy preliminary result, adapted from Carmona and Hu .
There exists some positive constant such that the solution of the KPZ equation with zero bare velocity,
Proof (see , Lemma 3.1) Let, for general forcing term,
Conditioning with respect to the terminal condition, , means that we average with respect to the law of the Brownian bridge from to (see e.g. ). Then, for ,
where . By construction, . Hence (by concavity of the log)
Taking the expectation with respect to the noise and using independence of from , together with space translation invariance, one gets the superadditive inequality,
On the other hand, by convexity of exp, . Fekete’s superadditive lemma allows us to conclude to the existence of some constant verifying (1.9). This is the constant whose existence is asserted in Main Theorem (see (0.6)). Furthermore, by Jensen’s inequality, , as observed already in , Prop. 1.4.
As mentioned in the Introduction, Carmona and Hu  actually prove the existence of a limit random variable a.s.-lim for the solution of the KPZ equation with velocity