There is no isolated interface edge in very supercritical percolation

TL;DR

This paper studies the structure of interfaces in supercritical percolation, showing that interface edges are closely clustered and providing estimates on their distribution, advancing understanding of connectivity in high-probability regimes.

Contribution

It introduces a coupling method for constrained and unconstrained percolation configurations, revealing the local clustering behavior of interface edges near the critical regime.

Findings

Interface edges are within logarithmic distance of each other or pivotal edges.

Provides estimates for the distribution of edges far from the interface.

Shows typical clustering behavior of interface edges in supercritical percolation.

Abstract

We consider the Bernoulli bond percolation model in a box $\Lambda$ (not necessarily parallel to the directions of the lattice) in the regime where the percolation parameter is close to $1$. We condition the configuration on the event that two opposite faces of the box are disconnected. We couple this configuration with an unconstrained percolation configuration. The interface edges are the edges which differ in the two configurations. We prove that, typically, each interface edge is within a distance of order $\ln|\Lambda|$ of another interface edge or of a pivotal edge. We derive an estimate for the law of an edge which is far from the cut and the interface edges.