Universal KPZ Fluctuations for Moderate Deviations of Random Walks in Random Environments

TL;DR

This work establishes universal KPZ fluctuations for moderate deviations in Random Walks in a Random Environment (RWRE). By a generalized replica/moment framework and a discrete Tanaka/doob–Meyer analysis of a two-point motion, the authors show that the tail probabilities of RWRE transitions converge to the stochastic heat equation (SHE) with multiplicative noise, whose log solves the KPZ equation. The convergence is proven at the level of the first two moments, which fixes the noise strength $D_0=rac{ extrm{Var}_ uig( extrm{E}^m{\xi}[Y]ig)}{(2D)^{3/2}}$, with $D=rac12 extstyle\sum_i ar{\xi}(i)i^2$ and a precise moderate-deviation scaling. The analysis relies on tilting the replica measures, relating moments to local times, and identifying a discrete-to-continuum local-time dilation via the invariant measure of the two-point gap process, thereby connecting microscopic environmental statistics to KPZ universality. The results generalize prior nearest-neighbor RWRE limits and offer a framework to study extreme diffusion statistics in random media, with potential experimental relevance for measuring diffusion coefficients from extreme-event data.

Abstract

The theory of diffusion seeks to describe the motion of particles in a chaotic environment. Classical theory models individual particles as independent random walkers, effectively forgetting that particles evolve together in the same environment. Random Walks in a Random Environment (RWRE) models treat the environment as a random space-time field that biases the motion of particles based on where they are in the environment. We provide a universality result for the moderate deviations of the transition probability of this model over a wide class of choices of random environments. In particular, we show the convergence of moments to those of the multiplicative noise stochastic heat equation (SHE), whose logarithm is the Kardar-Parisi-Zhang (KPZ) equation. The environment only filters into the scaling limit through one parameter, which depends explicitly on the statistical description of the environment. This forms the basis for our introduction, in arXiv:2406.17733, of the extreme diffusion coefficient.