Kardar-Parisi-Zhang equation and large deviations for random walks in weak random environments

TL;DR

This paper establishes a connection between random walks in weak random environments and the KPZ universality class by proving large deviation principles and convergence to the stochastic heat equation in 1+1 dimensions.

Contribution

It proves a sharp large deviation result for RWRE in weak disorder and demonstrates convergence to the SHE, linking RWRE behavior to KPZ universality.

Findings

Transition probabilities converge to SHE with multiplicative noise

Large deviation regime characterized by KPZ equation

Application to Beta RWRE model

Abstract

We consider the transition probabilities for random walks in $1+1$ dimensional space-time random environments (RWRE). For critically tuned weak disorder we prove a sharp large deviation result: after appropriate rescaling, the transition probabilities for the RWRE evaluated in the large deviation regime, converge to the solution to the stochastic heat equation (SHE) with multiplicative noise (the logarithm of which is the KPZ equation). We apply this to the exactly solvable Beta RWRE and additionally present a formal derivation of the convergence of certain moment formulas for that model to those for the SHE.