Lower large deviations for the maximal flow through a domain of ${\mathbb{R}}^d$ in first passage percolation

Abstract

We consider the standard first passage percolation model in the rescaled graph ${\mathbb{Z}}^d/n$ for $d\ge 2$, and a domain $\Omega$ of boundary $\Gamma$ in ${\mathbb{R}}^d$. Let ${\Gamma }^1$ and ${\Gamma }^2$ be two disjoint open subsets of $\Gamma$, representing the parts of $\Gamma$ through which some water can enter and escape from $\Omega$. We investigate the asymptotic behaviour of the flow ${\varphi }_n$ through a discrete version ${\Omega }_n$ of $\Omega$ between the corresponding discrete sets ${\Gamma }_n^1$ and ${\Gamma }_n^2$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the lower large deviations of ${\varphi }_n/n^{d-1}$ below a certain constant are of surface order.

23 pages, 8 figures