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

Abstract

Consider the first passage percolation on the d-dimensional lattice Zd with identical and independent weight distributions and the first passage time T. In this paper, we study the upper tail large deviations P(T(0,nx)>n(μ+ξ)), for ξ>0 and x≠0 with a time constant μ, for weights that satisfy a tail assumption P(τe>t)≍βexp(−αtr). When r≤1 (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as exp(−(2dαξr+o(1))n). When 1<r≤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 r=d is critical and logarithmic corrections appear. For r∈(1,d), we show that the large deviation event {T(0,nx)>n(μ+ξ)} is described by a localization of high weights around the endpoints. The picture changes for r≥d where the configuration is not anymore localized.