Coupling Invasion and First Passage Percolation

TL;DR

This paper establishes a coupling between invasion percolation and first passage percolation with log-uniform times, providing a new criterion for continuous phase transitions in bond percolation on lattices.

Contribution

It introduces a novel coupling method linking invasion and first passage percolation, offering an alternative perspective on phase transition criteria.

Findings

Coupling between invasion and first passage percolation with log-uniform times.

New equivalent condition for continuous phase transition in bond percolation.

Insight into the relationship between invasion percolation and phase transition behavior.

Abstract

It is well known that a continuous phase transition in Bernoulli bond percolation on the integer lattice is equivalent to a vanishing probability a vertex is invaded in invasion percolation. We provide a coupling between invasion percolation and first passage percolation with log-uniform passage times. This yields a new equivalent condition for a continuous phase transition in bond percolation.