MARK KAC SEMINAR

Planar random growth processes occur widely in the physical world. One of the most well-known, yet notoriously difficult, examples is diffusion-limited aggregation (DLA) which models mineral deposition. This process is usually initiated from a cluster containing a single ”seed” particle, which successive particles then attach themselves to. However, physicists have also studied DLA seeded on a line segment. One approach to mathematically modelling planar random growth seeded from a single particle is to take the seed particle to be the unit disk and to represent the randomly growing clusters as compositions of conformal mappings of the exterior unit disk. In 1998, Hastings and Levitov proposed a family of models using this approach, which includes a version of DLA. In this talk I will define a stationary version of the Hastings-Levitov model by composing conformal mappings in the upper half-plane. This is proposed as a candidate for off-lattice DLA seeded on the line. We analytically derive various properties of this model and show that they agree with numerical experiments for DLA in the physics literature.

This talk is based on joint works with Eviatar Procaccia and Amanda Turner.

In the first part of the talk I will review Toom's classical result about stability of trajectories of cellular automata. Informally, we say that a cellular automaton is stable if it does not completely lose memory of its initial state when subjected to noise. Using a contour argument Toom gave necessary and sufficient conditions for the cellular automaton to be stable. I will introduce an alternative construction of Toom contours that allows us to extend his method to more general models. In the second part of the talk I will discuss possible applications and limitations of this extension. I will derive stability results for certain continuous-time processes, as well as provide an alternative proof of the fact that subcritical bootstrap percolation models have a positive critical probability in any dimension. (Based on joint work with Ivailo Hartarsky, Jan Swart and Cristina Toninelli.)