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

Georg Menz
University of California, Los Angeles (UCLA)

The main focus of this presentation is the study of ergodicity of
Interacting Particle Systems (IPS). We present a simple time-scaling
lemma showing that rescaling 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 almost trivial observation has
non-trivial implications. It allows us to extend any result that does
not respect this invariance. As an application we deduce a new
criterion for the decay of correlation for IPS with alphabets of
arbitrary (finite) size using a straightforward recursive method and
extend it using the time-scaling lemma. Another application allows the
transfer of ergodicity from probabilistic cellular automata to IPS
under sufficient conditions. We close the presentation by discussing
the relevance of those observations to the positive rates conjecture.


This presentation is based on collaborative work with Maciej
Gluchowski from the University of Warsaw and Jacob Manaker from UCLA


Back to Geometry, Statistical Mechanics, and Integrability