Random-walk in Beta-distributed random environment

TL;DR

This paper introduces the Beta RWRE, an exactly solvable model of one-dimensional random walk in a Beta-distributed environment, and demonstrates Tracy-Widom fluctuations in its position distribution.

Contribution

It provides an exact solution for Beta RWRE, linking it to directed polymers and deriving Tracy-Widom limit theorems for its position distribution.

Findings

Proves second order cube-root scale corrections to large deviations.

Establishes Tracy-Widom distribution as the limit law for the walker's position.

Connects the model to maximum of correlated random variables and directed polymer limits.

Abstract

We introduce an exactly-solvable model of random walk in random environment that we call the Beta RWRE. This is a random walk in $\mathbb{Z}$ which performs nearest neighbour jumps with transition probabilities drawn according to the Beta distribution. We also describe a related directed polymer model, which is a limit of the $q$-Hahn interacting particle system. Using a Fredholm determinant representation for the quenched probability distribution function of the walker's position, we are able to prove second order cube-root scale corrections to the large deviation principle satisfied by the walker's position, with convergence to the Tracy-Widom distribution. We also show that this limit theorem can be interpreted in terms of the maximum of strongly correlated random variables: the positions of independent walkers in the same environment. The zero-temperature counterpart of the Beta RWRE can be studied in a parallel way. We also prove a Tracy-Widom limit theorem for this model.