Stochastic, Distributed Separation and Integration in Heterogeneous Particle Systems

We study a distributed system of interacting particles that move randomly over a lattice, where movement probabilities are governed by the affinity particles have for others of the same color (or type). Under what conditions will the particles separate into monochromatic clusters or integrate, indifferent to color? We present a distributed, stochastic algorithm that can achieve both separation and integration depending on a single parameter determining the strength of preference that particles have for being near others of the same color. Proving the correctness of this algorithm requires several new techniques including the cluster expansion from statistical physics; a variant of the bridging argument by Miracle, Pascoe, and Randall (RANDOM 2011); and the high-temperature expansion of the Ising model of ferromagnetism.

This is joint work with Sarah Cannon, Dana Randall, and Andréa Richa.