Large deviations in first-passage percolation

Abstract

Consider the standard first-passage percolation on ${\Z}^d$, $d\geq 2$. Denote by $\phi_{0,n}$ the face--face first-passage time in $[0,n]^d$. It is well known that \[ \lim_{n\rightarrow \infty} {\phi_{0,n}\over n}=\mu(F) \qquad \mbox{a.s. and in } L_1, \] where $F$ is the common distribution on each edge. In this paper we show that the upper and lower tails of $\phi_{0,n}$ are quite different when $\mu(F)>0$. More precisely, we can show that for small $\varepsilon>0$, there exist constants $\alpha(\varepsilon, F)$ and $\beta (\varepsilon, F)$ such that \[ \lim_{n\rightarrow\infty}{-1\over n} \log P \left( \phi_{0,n}\leq n(\mu -\varepsilon) \right) = \alpha (\varepsilon, F) \] and \[ \lim_{n\rightarrow\infty}{-1\over n^d} \log P \left(\phi_{0,n}\geq n(\mu +\varepsilon) \right)= \beta (\varepsilon, F). \]