Upper large deviations for the maximal flow in first passage percolation

Details Upper large deviations for the maximal flow in first passage percolation Upper large deviations for the maximal flow in first passage percolation

2006-07-11

arxiv.org/abs/math/0607253v2

Stochastic Processes and their Applications, Volume 117, Issue 9, Septembre 2007, Pages 1208-1233

Mathematics

Probability

27 pages, 4 figures; small changes of notations

Scientific paper

10.1016/j.spa.2006.12.007

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 $(\phi_{n^{d-1},h(n)} / n^{d-1}, n\in \mathbb{N})$.

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