On large deviation regimes for random media models

TL;DR

This paper investigates the asymmetric behavior of large deviations in random media models, specifically in first and last passage percolation times, revealing significant differences in deviations above and below the mean.

Contribution

It introduces robust methods to quantify and explain the asymmetry in large deviation behaviors for these percolation functionals.

Findings

Large deviations above the mean differ markedly from those below.

Developed techniques quantify the asymmetry in large deviation regimes.

Results enhance understanding of probabilistic behaviors in random media.

Abstract

The focus of this article is on the different behavior of large deviations of random subadditive functionals above the mean versus large deviations below the mean in two random media models. We consider the point-to-point first passage percolation time $a_n$ on $\mathbb{Z}^d$ and a last passage percolation time $Z_n$. For these functionals, we have $\lim_{n\to\infty}\frac{a_n}{n}=\nu$ and $\lim_{n\to\infty}\frac{Z_n}{n}=\mu$. Typically, the large deviations for such functionals exhibits a strong asymmetry, large deviations above the limiting value are radically different from large deviations below this quantity. We develop robust techniques to quantify and explain the differences.