Second order cubic corrections of large deviations for perturbed random walks*

We prove that the Beta random walk, introduced in [BC17] 2017, has cubic fluctuations from the large deviation principle of the GUE Tracy-Widom type for arbitrary values ?? > 0 and (3 > 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.

Automated summary

This study proves that Beta random walk exhibits cubic fluctuations for arbitrary parameter values and removes previous restrictions on their values. It also demonstrates the persistence of GUE Tracy-Widom fluctuations in the intermediate disorder regime.