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

TL;DR

This paper studies the probability of unusually low maximal flow in a first passage percolation model within a domain, showing these deviations decay at a surface order under certain regularity and capacity conditions.

Contribution

It establishes the asymptotic behavior of lower large deviations for maximal flow in a rescaled lattice domain, extending understanding of flow fluctuations in percolation models.

Findings

Lower large deviations decay at surface order.

Results depend on domain regularity and edge capacity distribution.

Provides conditions under which deviations are quantifiable.

Abstract

We consider the standard first passage percolation model in the rescaled graph $\mathbb{Z}^d/n$ for $d\geq 2$, 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 $\phi_n$ 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 lower large deviations of $\phi_n/ n^{d-1}$ below a certain constant are of surface order.