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

Abstract

Consider the first passage percolation on the d-dimensional lattice ${\mathbb{Z}}^{\mathit{d}}$ with identical and independent weight distributions and the first passage time T. In this paper, we study the upper tail large deviations $\mathbb{P}(\mathrm{T}(0,\mathit{n}\mathit{x})>\mathit{n}(\mathit{\mu }+\mathit{\xi }))$, for $\mathit{\xi }>0$ and $\mathit{x}\ne 0$ with a time constant μ, for weights that satisfy a tail assumption $\mathbb{P}({\mathit{\tau }}_{\mathit{e}}>\mathit{t})\asymp \mathit{\beta }exp(-\mathit{\alpha }{\mathit{t}}^{\mathit{r}})$. When $\mathit{r}\le 1$ (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as $exp(-(2\mathit{d}\mathit{\alpha }{\mathit{\xi }}^{\mathit{r}}+\mathit{o}(1))\mathit{n})$. When $1<\mathit{r}\le \mathit{d}$, we find that the rate function can be naturally described by a variational formula, called the discrete p-Capacity, and we study its asymptotics. The case $\mathit{r}=\mathit{d}$ is critical and logarithmic corrections appear. For $\mathit{r}\in (1,\mathit{d})$, we show that the large deviation event $\{\mathrm{T}(0,\mathit{n}\mathit{x})>\mathit{n}(\mathit{\mu }+\mathit{\xi })\}$ is described by a localization of high weights around the endpoints. The picture changes for $\mathit{r}\ge \mathit{d}$ where the configuration is not anymore localized.