Upper large deviations for the maximal flow through a domain of $\bolds{\mathbb{R}^d}$ 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 behavior 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 upper large deviations of ${\varphi }_n/n^{d-1}$ above a certain constant are of volume order, that is, decays exponentially fast with $n^d$. This article is part of a larger project in which the authors prove that this constant is the a.s. limit of ${\varphi }_n/n^{d-1}$.

Published in at http://dx.doi.org/10.1214/10-AAP732 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Limit theorems for maximum flows on a lattice
• Upper large deviations for the maximal flow in first passage percolation
• Capacitive flows on a 2D random net

Cited by in corpus (2)

• Maximal stream and minimal cutset for first passage percolation through a domain of ${\mathbb{R}}^d$
• Non-constant ground configurations in the disordered ferromagnet