The Ising Model on the Random Planar Causal Triangulation: Bounds on the Critical Line and Magnetization Properties

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 to 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.

Critical line of the Ising model on 2-dimensional CDT and its dual

“…In this paper we review some recent results, first appeared in Ref. 6, on the annealed coupling between the Ising model and the random causal triangulations of the plane. In particular, we show the existence of a critical line in the positive (β, µ) quarter plane, give a description of some of its analytical properties and provide bounds on the region where the critical line is located.…”

“…where the series on the right is finite (see Ref. 6) for all N and l whenever µ > 3 2 (log 2 + β). Therefore…”

Continuous spin models on annealed generalized random graphs

“…There is also an interest in annealed spin models on random graphs in the context of quantum gravity, see e.g. [36].…”

“…For the boundary case τ = k + 1 there are the following logarithmic corrections for β = 1/(k − 2): 36) and δ = k − 1:…”

Potts model coupled to random causal triangulations

“…The clasical example of such model is the two-state Potts model (Ising model) coupled to a CT introduced in [14]. For the Ising model existence of Gibbs measures and phase transitions has been recently proved (see [14], [15], [20], [25] and [28] for details). For the numerical results for the 3-state Potts model coupled to CTs we refer the reader to [16].…”