Large deviations and wandering exponent for random walk in a dynamic beta environment

TL;DR

This paper studies the behavior of a random walk in a dynamic beta environment, revealing its large deviation properties, the associated Doob transform, and its connection to KPZ universality with a wandering exponent of 2/3.

Contribution

It introduces a new analysis of the Doob transform for dynamic beta environments and links the walk's large deviations to KPZ universality class.

Findings

The transformed environment is correlated in time and i.i.d. in space.

The walk exhibits a wandering exponent of 2/3 under the averaged measure.

The harmonic function arises from a Busemann-type limit in a variational problem.

Abstract

Random walk in a dynamic i.i.d. beta random environment, conditioned to escape at an atypical velocity, converges to a Doob transform of the original walk. The Doob-transformed environment is correlated in time, i.i.d. in space, and its marginal density function is a product of a beta density and a hypergeometric function. Under its averaged distribution the transformed walk obeys the wandering exponent 2/3 that agrees with Kardar-Parisi-Zhang universality. The harmonic function in the Doob transform comes from a Busemann-type limit and appears as an extremal in a variational problem for the quenched large deviation rate function.