Random walk on the high-dimensional IIC

TL;DR

This paper investigates the asymptotic behavior of random walks on the incipient infinite cluster in high dimensions, establishing bounds on exit times and confirming the Alexander-Orbach conjecture for such clusters.

Contribution

It provides new bounds on effective resistance and demonstrates the Alexander-Orbach conjecture holds for high-dimensional IIC in both long-range and finite-range percolation.

Findings

Effective resistance bounds for IIC in high dimensions

Confirmation of the Alexander-Orbach conjecture for high-dimensional IIC

Distinct geometric properties of long-range versus finite-range clusters

Abstract

We study the asymptotic behavior the exit times of random walk from Euclidean balls around the origin of the incipient infinite cluster in a manner inspired by [26]. We do this by obtaining bounds on the effective resistance between the origin and the boundary of these Euclidean balls. We show that the geometric properties of long-range percolation clusters are significantly different from those of finite-range clusters. We also study the behavior of random walk on the backbone of the IIC and we prove that the Alexander-Orbach conjecture holds for the incipient infinite cluster in high dimensions, both for long-range percolation and for finite-range percolation.