Groups and Dynamics Seminar

Date: November 16, 2022

Time: 3:00PM - 4:00PM

Location: BLOC 506a

Speaker: Chris Shriver, University of Texas, Austin

Title: Non-equilibrium Gibbs states on a tree

Abstract: We consider two notions of statistical equilibrium for a probability-measure-preserving shift system: an “equilibrium state” maximizes a functional called the pressure while a “Gibbs state” satisfies a local equilibrium condition. Classical results of Dobrushin, Lanford, and Ruelle show that these notions are equivalent for Z^d systems, under some assumptions on the interaction, and the equivalence has been extended to arbitrary amenable groups. Barbieri and Meyerovitch have recently shown that one direction still holds for sofic groups: equilibrium states are always Gibbs. We will show that the converse fails in a nontrivial way using the example of the free boundary Ising state on an infinite regular tree (i.e. a free group): we show that for all temperatures below the uniqueness threshold this state is nonequilibrium over some sofic approximation, and below the reconstruction threshold it is nonequilibrium over every sofic approximation.