Random walk on nonnegative integers in beta distributed random environment

TL;DR

This paper analyzes a solvable model of a random walk on nonnegative integers in a beta-distributed random environment, revealing how boundary conditions influence local behavior and large deviations.

Contribution

It introduces an exactly solvable model with beta-distributed transition probabilities and derives formulas for moments and large-scale behavior analysis.

Findings

Boundary parameter $\\eta$ affects return probabilities.

Large deviation behavior matches that of the walk on $\mathbb{Z}$.

Provides explicit formulas for moments and asymptotics.

Abstract

We consider random walks on the nonnegative integers in a space-time dependent random environment. We assume that transition probabilities are given by independent $\mathrm{Beta}(\mu,\mu)$ distributed random variables, with a specific behaviour at the boundary, controlled by an extra parameter $\eta$. We show that this model is exactly solvable and prove a formula for the mixed moments of the random heat kernel. We then provide two formulas that allow us to study the large-scale behaviour. The first involves a Fredholm Pfaffian, which we use to prove a local limit theorem describing how the boundary parameter $\eta$ affects the return probabilities. The second is an explicit series of integrals, and we show that non-rigorous critical point asymptotics suggest that the large deviation behaviour of this half-space random walk in random environment is the same as for the analogous random walk on $\mathbb{Z}$.