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

Consider the first passage percolation on the d-dimensional lattice Z(d) 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(mu + xi)), for xi > 0 and x not equal 0 with a time constant mu, for weights that sat-isfy a tail assumption P(tau e > t) asymptotic to beta exp (-alpha t(r)). When r <= 1 (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as exp (-(2d alpha xi(r) + 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 is an element of (1, d), we show that the large de-viation event {T(0, nx) > n(mu + xi)} is described by a localization of high weights around the endpoints. The picture changes for r >= d where the con-figuration is not anymore localized.