Arm events in two-dimensional invasion percolation

TL;DR

This paper investigates the probabilities of arm events in two-dimensional invasion percolation and compares them to critical percolation, revealing conditions under which they are comparable or differ significantly, and establishing arm exponents for invasion percolation.

Contribution

It provides a detailed comparison of arm event probabilities between invasion and critical percolation, and establishes arm exponents for invasion percolation on the triangular lattice.

Findings

Arm probabilities are comparable for certain sequences, uniformly in n.

Arm probabilities differ by a power of n for other sequences.

Existence of arm exponents for invasion percolation with at least two open entries.

Abstract

We compare the probabilities of arm events in two-dimensional invasion percolation to those in critical percolation. Arm events are defined by the existence of a prescribed color sequence of invaded and non-invaded connections from the origin to distance n. We find that, for sequences of a particular form, arm probabilities in invasion percolation and critical percolation are comparable, uniformly in n, while they differ by a power of n for all others. A corollary of our results is the existence, on the triangular lattice, of arm exponents for invasion percolation, for any color sequence with at least two open (invaded) entries.