Size of a minimal cutset in supercritical first passage percolation

TL;DR

This paper investigates the size of minimal cutsets in supercritical first passage percolation on high-dimensional lattices, establishing almost sure convergence of the scaled minimal cutset size to a constant.

Contribution

It proves the almost sure convergence of the minimal cutset size, scaled by n^{d-1}, to a constant in supercritical first passage percolation.

Findings

Minimal cutset size scaled by n^{d-1} converges almost surely.

The limit of the scaled minimal cutset size is a positive constant.

The study extends understanding of flow and cut properties in supercritical percolation regimes.

Abstract

We consider the standard model of i.i.d. first passage percolation on Z^d given a distribution G on [0, +$\infty$] (including +$\infty$). We suppose that G({0}) > 1 -- p\_c(d), i.e., the edges of positive passage time are in the subcritical regime of percolation on Z^d. We consider a cylinder of basis an hyperrectangle of dimension d -- 1 whose sides have length n and of height h(n) with h(n) negligible compared to n (i.e., h(n)/n $\rightarrow$ 0 when n goes to infinity). We study the maximal flow from the top to the bottom of this cylinder. We already know that the maximal flow renormalized by n^(d--1) converges towards the flow constant which is null in the case G({0}) > 1 -- p\_c (d). The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut the top from the bottom of the cylinder. If we denote by $\psi$\_n the minimal cardinal of such a set of edges, we prove here that $\psi$\_n /n^(d--1) converges almost surely towards a constant.