The Scaling Limit of the KPZ Equation in Space Dimension 3 and Higher

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Abstract

We study in the present article the Kardar–Parisi–Zhang (KPZ) equation
$$\begin{aligned} \partial _t h(t,x)=\nu \Delta h(t,x)+\lambda |\nabla h(t,x)|^2 +\sqrt{D}\, \eta (t,x), \qquad (t,x)\in \mathbb {R}_+\times \mathbb {R}^d \end{aligned}$$
in \(d\ge 3\) dimensions in the perturbative regime, i.e. for \(\lambda >0\) small enough and a smooth, bounded, integrable initial condition \(h_0=h(t=0,\cdot )\). The forcing term \(\eta \) in the right-hand side is a regularized space-time white noise. The exponential of h—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 and Magnen (Commun Math Phys 162(1):85–121, 1994). Standard large deviation estimates for \(\eta \) 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 h 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 (\(\lambda =0\)) with renormalized coefficients \(\nu _{eff}=\nu +O(\lambda ^2),D_{eff}=D+O(\lambda ^2)\).

Keywords

KPZ equation Cole-Hopf transformation Directed polymer Constructive field theory Renormalization Cluster expansion 

Mathematics Subject Classification

35B50 35B51 35D40 35K55 35R60 35Q82 60H15 81T08 81T16 81T18 82C41 

Notes

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. J. Unterberger acknowledges the support of the ANR, via the ANR project ANR-16-CE40-0020-01.

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Authors and Affiliations

  1. 1.Laboratoire associé au CNRS UMR 7644, Centre de Physique ThéoriqueEcole PolytechniquePalaiseau CedexFrance
  2. 2.Laboratoire associé au CNRS UMR 7502, Institut Elie CartanUniversité de LorraineVandœuvre-lès-Nancy CedexFrance

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