Limit theorems for 2D invasion percolation

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 (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 (O(n)). We then show consequences of these limit theorems for the pond radii and outlet weights.