Upper tail large deviations for a class of distributions in First-passage percolation

TL;DR

This paper investigates the probabilities of large deviations in first passage percolation models, establishing exponential decay rates for certain distributions and highlighting cases where the rate function fails to exist.

Contribution

It provides precise exponential decay rates for upper tail large deviations in Eden growth models and extends results to stretched exponential distributions, also identifying distributions with no rate function.

Findings

Exponential decay rate of P(T(0,nx)>n(μ+ξ)) as exp(-(2dξ+o(1))n)

Extension of large deviation results to stretched exponential distributions

Existence of distributions with finite exponential moments where the rate function does not exist

Abstract

In this paper we consider the first passage percolation with identical and independent exponentially distributions, called the Eden growth model, and we study the upper tail large deviations for the first passage time ${\rm T}$. Our main results prove that for any $\xi>0$ and $x\neq 0$, $\mathbb{P}({\rm T}(0,nx)>n(\mu+\xi))$ decays as $\exp{(-(2d\xi +o(1))n)}$ with a time constant $\mu$ and a dimension $d$. Moreover, we extend the result to stretched exponential distributions. On the contrary, we construct a continuous distribution with a finite exponential moment where the rate function does not exist.