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

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 $x\mathrm{\ne }0$ 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 $1<r\le d$, 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.