The Ising model on the random planar causal triangulation: bounds on the critical line and magnetization properties

TL;DR

This paper studies the Ising model on random planar causal triangulations, providing bounds on the critical temperature and analyzing how boundary conditions affect magnetization, especially in the infinite volume limit.

Contribution

It establishes bounds on the critical line for the Ising model on causal triangulations and analyzes magnetization sensitivity to boundary conditions.

Findings

Magnetization of the central spin depends on boundary conditions in finite triangulations.

In the infinite volume limit, the magnetization vanishes at high temperatures.

Provides bounds on the critical line where the Gibbs measure is defined.

Abstract

We investigate a Gibbs (annealed) probability measure defined on Ising spin configurations on causal triangulations of the plane. We study the region where such measure can be defined and provide bounds on the boundary of this region (critical line). We prove that for any finite random triangulation the magnetization of the central spin is sensitive of the boundary conditions. Furthermore, we show that in the infinite volume limit, the magnetization of the central spin vanishes for values of the temperature high enough.