The size of a pond in 2D invasion percolation

TL;DR

This paper establishes that the probability of large ponds in 2D invasion percolation is asymptotically equivalent to the probability of large open clusters in critical Bernoulli percolation, including volume considerations.

Contribution

It proves that the probabilities of large ponds and large open clusters are of the same order, refining previous bounds and extending understanding of pond sizes in invasion percolation.

Findings

Probabilities of large ponds and large clusters are asymptotically equivalent.

Results apply to both pond radius and volume.

Extends previous bounds to exact order equivalence.

Abstract

We consider invasion percolation on the square lattice. It has been proved by van den Berg, Peres, Sidoravicius and Vares, that the probability that the radius of a so-called pond is larger than n, differs at most a factor of order log n from the probability that in critical Bernoulli percolation the radius of an open cluster is larger than n. We show that these two probabilities are, in fact, of the same order. Moreover, we prove an analogous result for the volume of a pond.