Random graph asymptotics on high-dimensional tori. II. Volume, diameter and mixing time

TL;DR

This paper establishes sharp bounds on the size, diameter, and mixing time of the largest clusters in critical percolation on high-dimensional tori, confirming they behave similarly to critical Erdős-Rényi graphs.

Contribution

It proves sharp lower bounds on cluster sizes, resolving a conjecture about boundary conditions, and extends results to multiple large clusters in high-dimensional percolation.

Findings

Sharp lower bounds on the largest cluster size.

Boundary condition effects are clarified, confirming Erdős-Rényi behavior.

Bounds on diameter and mixing time of clusters are established.

Abstract

For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (2007). This improvement finally settles a conjecture by Aizenman (1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further prove that the volume bounds apply also to any finite number of the largest clusters. The main conclusion of the paper is that the behavior of critical percolation on the high-dimensional torus is the same as for critical Erdos-Renyi random graphs. In this updated version we incorporate an erratum to be published in a forthcoming issue of Probab. Theory Relat. Fields. This results in a modification of Theorem 1.2 as well as Proposition 3.1.