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

TL;DR

This paper studies the probability of unusually high maximal flow in a first passage percolation model, showing that such large deviations decay exponentially with the volume of the domain, under certain regularity and capacity conditions.

Contribution

It establishes the exponential decay rate of upper large deviations for the maximal flow in first passage percolation, extending understanding of flow behavior in high-dimensional random media.

Findings

Large deviations decay exponentially with the domain volume

The constant limit of the scaled maximal flow is identified

Conditions on domain regularity and edge capacities are specified

Abstract

We consider the standard first passage percolation model in the rescaled graph $\mathbb {Z}^d/n$ for $d\geq2$ 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 $\phi_n$ through a discrete version $\Omega_n$ of $\Omega$ between the corresponding discrete sets $\Gamma ^1_n$ and $\Gamma ^2_n$. 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 $\phi_n/n^{d-1}$ 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 $\phi_n/n^{d-1}$.