Exponential growth of ponds in invasion percolation on regular trees

TL;DR

This paper investigates the exponential growth of ponds in invasion percolation on regular trees, providing probabilistic limit theorems and tail asymptotics linked to critical percolation clusters.

Contribution

It establishes the exponential growth rate of ponds and derives limit theorems and tail asymptotics, connecting invasion percolation to critical percolation theory.

Findings

Ponds grow exponentially in invasion percolation on regular trees.

Law of large numbers, CLT, and large deviation principles are proved.

Tail asymptotics relate to critical percolation clusters with a logarithmic correction.

Abstract

In invasion percolation, the edges of successively maximal weight (the outlets) divide the invasion cluster into a chain of ponds separated by outlets. On the regular tree, the ponds are shown to grow exponentially, with law of large numbers, central limit theorem and large deviation results. The tail asymptotics for a fixed pond are also studied and are shown to be related to the asymptotics of a critical percolation cluster, with a logarithmic correction.