Examples of nonpolygonal limit shapes in i.i.d. first-passage percolation and infinite coexistence in spatial growth models

We construct an edge-weight distribution for i.i.d. first-passage percolation on $\mathbb{Z}^2$ whose limit shape is not a polygon and whose extreme points are arbitrarily dense in the boundary. Consequently, the associated Richardson-type growth model can support coexistence of a countably infinite number of distinct species, and the graph of infection has infinitely many ends.Comment: Published in at http://dx.doi.org/10.1214/12-AAP864 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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 $({\mathbf{O}}(n))$ 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.Comment: 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. Wahrsc