Relations between invasion percolation and critical percolation in two dimensions

TL;DR

This paper explores the relationship between invasion percolation and critical percolation in two dimensions, revealing both similarities in asymptotic behaviors and key differences in cluster structure and measure properties.

Contribution

It establishes the asymptotic equivalence of the $k$-point function for ponds with critical clusters and demonstrates the measure singularity between invasion percolation and incipient infinite clusters.

Findings

The $k$-point function of ponds matches the critical cluster probability asymptotically.

There are infinitely many ponds with large disjoint $p_c$-open clusters.

The decay rate of the radius distribution of the $k$th pond differs from that of the critical cluster.

Abstract

We study invasion percolation in two dimensions. We compare connectivity properties of the origin's invaded region to those of (a) the critical percolation cluster of the origin and (b) the incipient infinite cluster. To exhibit similarities, we show that for any $k\geq1$, the $k$-point function of the first so-called pond has the same asymptotic behavior as the probability that $k$ points are in the critical cluster of the origin. More prominent, though, are the differences. We show that there are infinitely many ponds that contain many large disjoint $p_c$-open clusters. Further, for $k>1$, we compute the exact decay rate of the distribution of the radius of the $k$th pond and see that it differs from that of the radius of the critical cluster of the origin. We finish by showing that the invasion percolation measure and the incipient infinite cluster measure are mutually singular.