Probing the large deviations for the Beta random walk in random medium

TL;DR

This paper investigates the large deviations of a Beta random walk in a random medium, deriving the rate function, confirming its universality with a continuum model, and validating predictions through extensive numerical simulations.

Contribution

It provides the first detailed analysis of large deviations for the Beta random walk, linking it to continuum models and confirming the predicted phase transition numerically.

Findings

Rate function matches that of the continuum model after rescaling.

Numerical simulations agree with analytical predictions.

First-order transition observed in numerics as predicted.

Abstract

We consider a discrete-time random walk on a one-dimensional lattice with space and time-dependent random jump probabilities, known as the Beta random walk. We are interested in the probability that, for a given realization of the jump probabilities (a sample), a walker starting at the origin at time $t=0$ is at position beyond $\xi \sqrt{T/2}$ at time $T$. This probability fluctuates from sample to sample and we study the large-deviation rate function which characterizes the tails of its distribution at large time $T \gg 1$. It is argued that, up to a simple rescaling, this rate function is identical to the one recently obtained exactly by two of the authors for the continuum version of the model. That continuum model also appears in the macroscopic fluctuation theory of a class of lattice gases, e.g. in the so-called KMP model of heat transfer. An extensive numerical simulation of the Beta random walk, based on an importance sampling algorithm, is found in good agreement with the detailed analytical predictions. A first-order transition in the tilted measure, predicted to occur in the continuum model, is also observed in the numerics.