The variance of the graph distance in the infinite cluster of percolation is sublinear

TL;DR

This paper proves that in supercritical bond percolation on bZ^d, the variance of the graph distance between two points in the infinite cluster grows slower than linearly, extending previous results in related models.

Contribution

It establishes the sublinear growth of variance of graph distances in the infinite cluster for supercritical percolation, using recent advanced techniques.

Findings

Variance of graph distance is sublinear in the infinite cluster

Extends previous variance results to percolation models without moment conditions

Utilizes recent techniques of Cerf and Dembin

Abstract

We consider the standard model of i.i.d. bond percolation on $\mathbb Z^d$ of parameter $p$. When $p>p_c$, there exists almost surely a unique infinite cluster $\mathcal C_p$. Using the recent techniques of Cerf and Dembin, we prove that the variance of the graph distance in $\mathcal C_p$ between two points of $\mathcal C_p$ is sublinear. The main result extends the works of Benjamini, Kalai and Schramm, Benaim and Rossignol and Damron, Hanson and Sosoe for the study of the variance of passage times in first passage percolation without moment conditions on the edge-weight distribution.