The scaling limit of the KPZ equation in space dimension 3 and higher

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Description:	Unterberger, J Thursday 25th October 2018 - 10:00 to 11:00


Created:	2018-10-29 10:16
Collection:	Scaling limits, rough paths, quantum field theory
Publisher:	Isaac Newton Institute
Copyright:	Unterberger, J
Language:	eng (English)


Abstract:	We study in the present article the Kardar-Parisi-Zhang (KPZ) equation ∂th(t,x)=ν\Delh(t,x)+λ\|∇h(t,x)\|2+D−−√η(t,x),(t,x)∈R+×Rd in d≥3 dimensions in the perturbative regime, i.e. for λ>0 small enough and a smooth, bounded, integrable initial condition h0=h(t=0,⋅). The forcing term η 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-Magnen. Standard large deviation estimates for η 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 (λ=0) with renormalized coefficients νeff=ν+O(λ2),Deff=D+O(λ2). This is joint work with J. Magnen.

Abstract:

We study in the present article the Kardar-Parisi-Zhang (KPZ) equation
∂th(t,x)=ν\Delh(t,x)+λ|∇h(t,x)|2+D−−√η(t,x),(t,x)∈R+×Rd
in d≥3 dimensions in the perturbative regime, i.e. for λ>0 small enough and a smooth, bounded, integrable initial condition h0=h(t=0,⋅). The forcing term η 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-Magnen. Standard large deviation estimates for η 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 (λ=0) with renormalized coefficients νeff=ν+O(λ2),Deff=D+O(λ2). This is joint work with J. Magnen.

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