Upper large deviations for the maximal flow in first passage percolation

Abstract

We consider the standard first passage percolation in $\mathbb{Z}^{d}$ for $d\geq 2$ and we denote by $\phi_{n^{d-1},h(n)}$ the maximal flow through the cylinder $]0,n]^{d-1} \times ]0,h(n)]$ from its bottom to its top. Kesten proved a law of large numbers for the maximal flow in dimension three: under some assumptions, $\phi_{n^{d-1},h(n)} / n^{d-1}$ converges towards a constant $\nu$. We look now at the probability that $\phi_{n^{d-1},h(n)} / n^{d-1}$ is greater than $\nu + \epsilon$ for some $\epsilon >0$, and we show under some assumptions that this probability decays exponentially fast with the volume of the cylinder. Moreover, we prove a large deviations principle for the sequence $(\phi_{n^{d-1},h(n)} / n^{d-1}, n\in \mathbb{N})$.