Time-Scaling, Ergodicity, and Covariance Decay of Interacting Particle Systems

Abstract

The main focus of this article is the study of ergodicity of Interacting Particle Systems (IPS). We present a simple lemma showing that scaling time is equivalent to taking the convex combination of the transition matrix of the IPS with the identity. As a consequence, the ergodic properties of IPS are invariant under this transformation. Surprisingly, this simple observation has non-trivial implications: It allows to extend any result that does not respect this invariance, which we demonstrate with examples. Additionally, we develop a recursive method to deduce decay of correlations for IPS with alphabets of arbitrary (finite) size, and apply the Time-Scaling Lemma to that as well. As an application of this new criterion we show that certain one-dimensional IPS are ergodic answering an open question of Toom et al.