Limit theorems for 2D invasion percolation

TL;DR

This paper establishes limit theorems and variance estimates for quantities related to invasion percolation in two dimensions, revealing the statistical behavior of outlets and pond sizes.

Contribution

It introduces new properties of outlet sequences, including renewal and mixing behaviors, and proves central limit theorems and laws of large numbers for these quantities.

Findings

Sequence of outlet variables exhibits renewal structure and exponential mixing.

Central limit theorem holds for outlet counts in growing boxes.

Strong law of large numbers applies to outlet and pond size measures.

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.