Phase transition for the Ising model on the Critical Lorentzian triangulation

TL;DR

This paper investigates the phase transition of the Ising model on a Lorentzian triangulation, establishing the uniqueness of Gibbs measure at high temperatures and multiple measures at low temperatures, with a focus on the critical temperature.

Contribution

It proves the existence of a phase transition for the Ising model on Lorentzian triangulations, identifying the critical temperature and analyzing Gibbs measures.

Findings

Uniqueness of Gibbs measure at high temperature

Multiple Gibbs measures at low temperature

Critical temperature is almost surely constant

Abstract

Ising model without external field on an infinite Lorentzian triangulation sampled from the uniform distribution is considered. We prove uniqueness of the Gibbs measure in the high temperature region and coexistence of at least two Gibbs measures at low temperature. The proofs are based on the disagreement percolation method and on a variant of Peierls method. The critical temperature is shown to be constant a.s.