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

Abstract

Consider first passage percolation with identical and independent weight distributions and first passage time $\mathrm{T}$. In this paper, we study the upper tail large deviations $\mathbb{P}(\mathrm{T}(0,nx)>n(\mu +\xi ))$, for $\xi >0$ and $x\ne 0$ with a time constant $\mu$ and a dimension $d$, for weights that satisfy a tail assumption ${\beta }_1\exp (-\alpha t^r)\le \mathbb{P}({\tau }_e>t)\le {\beta }_2\exp (-\alpha t^r).$ When $r\le 1$ (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 $1<r\le d$, 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 $\mathrm{T}(0,nx)>n(\mu +\xi )$ is described by a localization of high weights around the origin. The picture changes for $r\ge d$ where the configuration is not anymore localized.

This preprint supersedes arXiv:1912.13212. 36 pages, 2 figures, v2

References in corpus (1)

• Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in random environment