Limit theorems for 2D invasion percolation

Abstract

We prove limit theorems and variance estimates for quantities related to ponds and outlets for 2D invasion percolation. We first exhibit several properties of a sequence $(\mathbf{O}(n))$ of outlet variables, the $n$th of which gives the number of outlets in the box centered at the origin of side length $2^n$. The most important of these properties describes the sequence's renewal structure and exponentially fast mixing behavior. We use these to prove a central limit theorem and strong law of large numbers for $(\mathbf{O}(n))$. We then show consequences of these limit theorems for the pond radii and outlet weights.

Published in at http://dx.doi.org/10.1214/10-AOP641 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org). Note: the statement of Lemma 3 (which is Theorem 2.1 in Withers (1981)) should include the condition liminf_n b_n^2/n > 0, which is valid in our setting. See the corrigendum to Theorem 2.1 in Withers (1983) in Z. Wahrsch

References in corpus (5)

• Relations between invasion percolation and critical percolation in two dimensions
• Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters
• Invasion percolation on regular trees
• Exponential growth of ponds in invasion percolation on regular trees
• Near-critical percolation in two dimensions

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• Minimal spanning trees at the percolation threshold: a numerical calculation
• Some toy models of self-organized criticality in percolation