Existence and continuity of the flow constant in first passage percolation

TL;DR

This paper investigates the existence, convergence, and continuity of the flow constant in first passage percolation on Z^d, focusing on (d-1)-dimensional minimal surfaces and establishing results under minimal assumptions on the distribution.

Contribution

It proves the existence of the flow constant without moment assumptions, shows convergence of maximal flows, and demonstrates its continuity with respect to the distribution G.

Findings

Flow constant exists under minimal conditions

Maximal flows converge to the flow constant

Flow constant is continuous with respect to G

Abstract

We consider the model of i.i.d. first passage percolation on Z^d, where we associate with the edges of the graph a family of i.i.d. random variables with common distribution G on [0, +$\infty$] (including +$\infty$). Whereas the time constant is associated to the study of 1-dimensional paths with minimal weight, namely geodesics, the flow constant is associated to the study of (d--1)-dimensional surfaces with minimal weight. In this article, we investigate the existence of the flow constant under the only hypothesis that G({+$\infty$}) < p c (d) (in particular without any moment assumption), the convergence of some natural maximal flows towards this constant, and the continuity of this constant with regard to the distribution G.