A Note About Critical Percolation on Finite Graphs

Abstract

In this note we study the geometry of the largest component \(\mathcal {C}_{1}\) of critical percolation on a finite graph G which satisfies the finite triangle condition, defined by Borgs et al. (Random Struct. Algorithms 27:137–184, 2005). 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. By Borgs et al. (Ann. Probab. 33:1886–1944, 2005), our results apply to critical percolation on several high-dimensional finite graphs such as the finite torus \(\mathbb{Z}_{n}^{d}\) (with d large and n→∞) and the Hamming cube {0,1}n.