A variational formula for large deviations in Firstpassage percolation under tail estimates
Abstract
Consider first passage percolation with identical and independent weight distributions and first passage time ${\rm T}$. In this paper, we study the upper tail large deviations $\mathbb{P}({\rm T}(0,nx)>n(\mu+\xi))$, for $\xi>0$ and $x\neq 0$ 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 wellknown Eden growth model), we show that the upper tail large deviation decays as $\exp{((2d\xi +o(1))n)}$. When $1< r\leq d$, we find that the rate function can be naturally described by a variational formula, called the discrete pCapacity, and we study its asymptotics. For $r<d$, we show that the large deviation event ${\rm T}(0,nx)>n(\mu+\xi)$ is described by a localization of high weights around the origin. The picture changes for $r\geq d$ where the configuration is not anymore localized.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 arXiv:
 arXiv:2101.08113
 Bibcode:
 2021arXiv210108113C
 Keywords:

 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 This preprint supersedes arXiv:1912.13212. 36 pages, 2 figures, v2