A variational formula for large deviations in First-passage percolation under tail estimates

TL;DR

This paper investigates the large deviation probabilities for first-passage percolation with tail estimates on weights, deriving a variational formula for the rate function and analyzing different regimes based on tail decay parameter r.

Contribution

It introduces a variational formula for the large deviation rate function in first-passage percolation under tail estimates, extending understanding across different tail decay regimes.

Findings

For r ≤ 1, large deviations decay exponentially with rate ~2dξn.

For 1 < r ≤ d, the rate function is characterized by a variational formula called the discrete p-Capacity.

For r < d, large deviations involve localization of high weights near the origin.

Abstract

Consider first passage percolation with identical and independent weight distributions and first passage time ${\rm T}$. In this paper, we study the upper tail large deviations $\mathbb{P}({\rm T}(0,nx)>n(\mu+\xi))$, for $\xi>0$ and $x\neq 0$ with a time constant $\mu$ and a dimension $d$, for weights that satisfy a tail assumption $ \beta_1\exp{(-\alpha t^r)}\leq \mathbb P(\tau_e>t)\leq \beta_2\exp{(-\alpha t^r)}.$ When $r\leq 1$ (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as $\exp{(-(2d\xi +o(1))n)}$. When $1< r\leq d$, we find that the rate function can be naturally described by a variational formula, called the discrete p-Capacity, and we study its asymptotics. For $r<d$, we show that the large deviation event ${\rm T}(0,nx)>n(\mu+\xi)$ is described by a localization of high weights around the origin. The picture changes for $r\geq d$ where the configuration is not anymore localized.