A note about critical percolation on finite graphs

TL;DR

This paper investigates the geometric properties of the largest component in critical percolation on finite graphs satisfying the finite triangle condition, revealing its size, diameter, and mixing time.

Contribution

It establishes the diameter and mixing time of the largest component at criticality, extending understanding to high-dimensional finite graphs.

Findings

Largest component size is n^{2/3}

Diameter of the component is n^{1/3}

Random walk mixing time is proportional to n

Abstract

In this note we study the geometry of the largest component C_1 of critical percolation on a finite graph G which satisfies the finite triangle condition, defined by Borgs et al. There it is shown that this component is of size n^{2/3}, and here we show that its diameter is n^{1/3} and that the simple random walk takes n steps to mix on it. Our results apply to critical percolation on several high-dimensional finite graphs such as the finite torus Z_n^d (with d large and n tending to infinity) and the Hamming cube {0,1}^n.