Gibbs-non-Gibbs properties for evolving Ising models on trees

TL;DR

This paper investigates how Gibbs measures on Cayley trees evolve under infinite-temperature Glauber dynamics, revealing complex transition behaviors between Gibbsian and non-Gibbsian regimes for different states.

Contribution

It characterizes the distinct non-Gibbsian transition phenomena for intermediate, plus, and minus states under dynamic evolution on trees.

Findings

Intermediate state becomes non-Gibbsian with all configurations bad at large times

Plus and minus states exhibit a Gibbsian to non-Gibbsian and back transition

Two transitions occur for each state, with different behaviors for intermediate versus plus/minus states

Abstract

In this paper we study homogeneous Gibbs measures on a Cayley tree, subjected to an infinite-temperature Glauber evolution, and consider their (non-)Gibbsian properties. We show that the intermediate Gibbs state (which in zero field is the free-boundary-condition Gibbs state) behaves different from the plus and the minus state. E.g. at large times, all configurations are bad for the intermediate state, whereas the plus configuration never is bad for the plus state. Moreover, we show that for each state there are two transitions. For the intermediate state there is a transition from a Gibbsian regime to a non-Gibbsian regime where some, but not all configurations are bad, and a second one to a regime where all configurations are bad. For the plus and minus state, the two transitions are from a Gibbsian regime to a non-Gibbsian one and then back to a Gibbsian regime again.