The stationary distribution of a continuous-time Markov chain is generally a complicated function of its transition rates. However, if the rates are i.i.d. random variables whose common distribution satisfies certain tail conditions, then the stationary distribution is essentially a simple function of the exit rates out of each state. This is the main result of a new preprint with Frank den Hollander and Dana Randall, which generalizes and makes a precise prediction by Chvykov et al. (2021) and settles a question raised by Bordenave, Caputo, and Chafaï (2012) under certain assumptions.

A new preprint with Shunhao Oh and Dana Randall explains how to program simple computational “particles” of different types to self-organize into corresponding spatial regions that have a given hierarchical structure. The size, shape, and hierarchy of the regions are determined by the particles’ densities and affinities for particles of different types. Proving that the algorithm works requires new techniques for analyzing the Gibbs distributions of fixed-magnetization models from equilibrium statistical mechanics.

Dana Randall and I generalize and prove the main claims of an emerging physical theory of nonequilibrium self-organization, called rattling theory, by interpreting its main claims in terms of Markov processes. The result is a principle that relates a local aspect of the dynamics to the global steady state. We apply it to random walks on random graphs, spin-glass dynamics, and models of ant colonies.