Dana Randall and I find thresholds in the local–global correlations exhibited by the steady-state distributions of a particle system and the dynamics of the Sherrington–Kirkpatrick spin glass.

In a new paper, Dana Randall and I identify conditions under which reaction kinetics on disordered energy landscapes exhibit high local–global correlation.

Siobhan Roberts wrote an article about the work that Frank den Hollander, Dana Randall, and I did together at SLMath.

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.