Stochastic Threshold Growth

Consider the following random model for a growing droplet in the plane. The model has 3 parameters: an update probability $p$, a threshold $\theta$ and a neighborhood of the origin $N$. This neighborhood is a finite symmetric subset of the 2-dimensional integer lattices, and the neighborhood of an arbitrary $x$ in the lattice is $x+N$. The droplet is a random sequence $ A_0 \subseteq A_1 \subseteq \dotsm $ of subsets of the two dimensional integer lattice generated as follows: a point $x$ joins $A_{i+1}$ with probability $p$ if the intersection of $A_i$ and the neighborhood of $x$ is at least $\theta $. This process, known as stochastic threshold growth, generalizes Eden's model, Richardson's rule and discrete threshold growth. It is known that droplets generated by Eden's model, Richardson's rule or discrete threshold growth converge to limiting shapes. In this talk, we discuss a combinatorial lemma that is central to the proof that limiting shapes exist for stochastic threshold growth.