The shape of large balls in highly supercritical percolation

TL;DR

This paper studies the geometric shape of large clusters in highly supercritical percolation, revealing they asymptotically resemble parabola arcs near the axes, by linking percolation distances to TASEP dynamics.

Contribution

It establishes a novel connection between percolation cluster geometry and TASEP, providing new asymptotic shape results near the supercritical threshold.

Findings

Large percolation balls are asymptotically parabola-shaped near axes

Connection between percolation distances and TASEP dynamics is demonstrated

Shape results hold as the percolation parameter approaches one

Abstract

We exploit a connection between distances in the infinite percolation cluster, when the parameter is close to one, and the discrete-time TASEP on $\mathbb{Z}$. This shows that when the parameter goes to one, large balls in the cluster are asymptotically shaped near the axes like arcs of parabola.