Title: Neighborhood growth models with one-dimensional nucleation

Abstract: Neighborhood growth cellular automata were introduced over 40 years ago as easy to describe models that exhibit the complex phenomena of nucleation and metastability. The most well-known of these is the threshold-2 growth model on the 2d integer lattice, wherein an initially occupied set of vertices is iteratively enlarged by occupying all vertices with at least two occupied neighbors. If the initially occupied set of vertices is chosen by randomly including each vertex independently with small probability p>0, then all vertices eventually become occupied, but it typically takes exponentially long in (1/p) for the origin to become occupied due to the rarity of nucleation sites. By contrast, we study certain “supercritical” growth rules that have +-shaped neighborhoods, and where the first occupation time of the origin is polynomial in (1/p). These rules exhibit one-dimensional nucleation in the sense that lines begin to grow before forming two-dimensional nuclei. In many cases, we show that the first occupation time is $p^{-a+o(1)}$ for an explicit constant $a$ depending on the rule. In one case, we establish a logarithmic correction to the polynomial passage time due to a growth trajectory that resembles a branching process, while in other cases the first occupation time is of pure polynomial order $p^{-a}$. Based on joint work with Daniel Blanquicett, Janko Gravner and Luke Wilson.

Share this Event