The maximal flow from a compact convex subset to infinity in first passage percolation on Z^d

TL;DR

This paper proves that in first passage percolation on Z^d, the maximal flow from a scaled convex set to infinity converges to a deterministic capacity depending on the boundary, extending previous 2D results to higher dimensions.

Contribution

It establishes the almost sure convergence of the rescaled maximal flow from a convex set to infinity in higher dimensions, confirming a conjecture for dimensions greater than two.

Findings

Rescaled maximal flow converges to a deterministic boundary-dependent capacity.

The capacity is expressed as an integral over the boundary of the convex set.

Extension of 2D results to higher dimensions in first passage percolation.

Abstract

We consider the standard first passage percolation model on Z^d with a distribution G on R+ that admits an exponential moment. We study the maximal flow between a compact convex subset A of R^d and infinity. The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut A from infinity. We prove that the rescaled maximal flow between nA and infinity $\phi$(nA)/n^ (d--1) almost surely converges towards a deterministic constant depending on A. This constant corresponds to the capacity of the boundary $\partial$A of A and is the integral of a deterministic function over $\partial$A. This result was shown in dimension 2 and conjectured for higher dimensions by Garet in [6].