Statistical Physics Models Governed by Diffusion

Abstract

In this article we consider two probability models: stationary diffusion limited aggregation (SDLA) and finitary random interlacements (FRI). SDLA is a stochastic process on the upper half planar lattice, growing from an infinite line, with local growth rate proportional to stationary harmonic measure. We first prove that stationary harmonic measure of an infinite set in the upper planar lattice can be represented as the proper scaling limit of the classical harmonic measure of truncations of the infinite set. Then we construct an infinite SDLA that is ergodic with respect to left-right integer translation. For FRI, we prove a phase transition in the connectivity of FRI FI^{u,T} on Z^d with respect to the average stopping time T .