Upper large deviations for the maximal flow in first passage percolation

Abstract

We consider the standard first passage percolation in ${\mathbb{Z}}^d$ for $d\ge 2$ and we denote by ${\varphi }_{n^{d-1},h(n)}$ 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, ${\varphi }_{n^{d-1},h(n)}/n^{d-1}$ converges towards a constant $\nu$. We look now at the probability that ${\varphi }_{n^{d-1},h(n)}/n^{d-1}$ 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 $({\varphi }_{n^{d-1},h(n)}/n^{d-1},n\in \mathbb{N})$.

27 pages, 4 figures; small changes of notations

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