Large deviation principle for the streams and the maximal flow in first passage percolation

TL;DR

This paper establishes a large deviation principle for the maximal flow and streams in first passage percolation models, providing a probabilistic understanding of rare flow configurations in a rescaled lattice domain.

Contribution

It derives a large deviation principle for the maximal stream in first passage percolation, extending the law of large numbers to quantify rare events.

Findings

Large deviation principle for maximal streams established

Rate function for upper large deviations derived

Contraction principle applied to maximal flow

Abstract

We consider the standard first passage percolation model in the rescaled lattice $\mathbb{Z}^d$ for $d\geq 2$ and a bounded domain $\Omega$ in $\mathbb R ^d$. We denote by $\Gamma^1$ and $\Gamma^2$ two disjoint subsets of $\partial \Omega$ representing respectively the source and the sink, i.e., where the water can enter in $\Omega$ and escape from $\Omega$. A maximal stream is a vector measure $\overrightarrow{\mu}_n^{max}$ that describes how the maximal amount of fluid can enter through $\Gamma^1$ and spreads in $\Omega$. Under some assumptions on $\Omega$ and $G$, we already know a law of large number for $\overrightarrow{\mu}_n^{max}$. The sequence $(\overrightarrow{\mu}_n^{max})_{n\geq 1} $ converges almost surely to the set of solutions of a continuous deterministic problem of maximal stream in an anisotropic network. We aim here to derive a large deviation principle for streams and deduce by contraction principle the existence of a rate function for the upper large deviations of the maximal flow in $\Omega$.