Distances in the highly supercritical percolation cluster

TL;DR

This paper investigates the asymptotic behavior of distances in the supercritical percolation cluster, revealing a specific factor increase related to the percolation parameter, and introduces a novel link with TASEP.

Contribution

It establishes a new connection between percolation distances and TASEP, providing a precise asymptotic formula for distance growth as p approaches 1.

Findings

Distances are asymptotically increased by a factor depending on p.

A new connection with TASEP is demonstrated.

The proof introduces novel techniques linking percolation and exclusion processes.

Abstract

On the supercritical percolation cluster with parameter p, the distances between two distant points of the axis are asymptotically increased by a factor 1+(1-p)/2+o(1-p) with respect to the usual distance. The proof is based on an apparently new connection with the TASEP (totally asymmetric simple exclusion process).