Second order cubic corrections of large deviations for perturbed random walks

TL;DR

This paper establishes second order cubic fluctuations for Beta random walks and confirms GUE Tracy-Widom fluctuations in intermediate disorder regimes, extending previous results to broader parameter ranges.

Contribution

It removes restrictions on Beta distribution parameters and proves GUE Tracy-Widom fluctuations for a wider class of random walks and environments.

Findings

Beta random walk exhibits second order cubic fluctuations.

GUE Tracy-Widom fluctuations hold in intermediate disorder regimes.

Fluctuations also appear in certain random walks in space-time random environments.

Abstract

We prove that the Beta random walk has second order cubic fluctuations from the large deviation principle of the GUE Tracy-Widom type for arbitrary values $\upalpha>0$ and $\upbeta>0$ of the parameters of the Beta distribution, removing previous restrictions on their values. Furthermore, we prove that the GUE Tracy-Widom fluctuations still hold in the intermediate disorder regime. We also show that any random walk in space-time random environment that matches certain moments with the Beta random walk also has GUE Tracy-Widom fluctuations in the intermediate disorder regime. As a corollary we show the emergence of GUE Tracy-Widom fluctuations from the large deviation principle for trajectories ending at boundary points for random walks in space (time-independent) i.i.d. Dirichlet random environment in dimension $d=2$ for a class of asymptotic behavior of the parameters.