Bounds on the critical line via transfer matrix methods for an Ising model coupled to causal dynamical triangulations

Abstract

We introduce a transfer matrix formalism for the (annealed) Ising model coupled to two-dimensional causal dynamical triangulations. Using the Krein-Rutman theory of positivity preserving operators we study several properties of the emerging transfer matrix. In particular, we determine regions in the quadrant of parameters β, µ > 0 where the infinite-volume free energy converges, yielding results on the convergence and asymptotic properties of the partition function and the Gibbs measure.2000 MSC. 60F05, 60J60, 60J80.

Potts model coupled to random causal triangulations

“…As a byproduct, the Theorem 2 serves to find lower and upper bounds for the infinite-volume free energy. Moreover, in the case of 2-state Potts model (Ising model), Theorem 2 extends earlier results from [25], [20] and improves the numerical approximation of the curve in high temperature given in [14]. In aditional, this approach allows to get a better aproximation of the critical curve and check the asymptotic behavior of the critical curve given in [14], and it say that critical curve is asymptotic to 3 2 β + ln 2.…”

“…Finally, we give a short of the Edwards-Sokal coupling. We refer to [19], [27], [20], for more details. We attempt at establishing regions where the infinite-volume free energy converges, yielding results on the con-vergence and asymptotic properties of the partition function and the Gibbs measure.…”

“…That heuristic result was proved in [20]. The following properties hold and will be utilized to prove the main results.…”

“…In this section we only consider Ising model on dual causal triangulations in order to compare our results with the previous results about the Ising model coupled dual causal triangulations present in [20]. That early work utilize the transfer matrix method in order to provide a curve µ * = ψ(β * ) (blue line in Figure 5) that satisfies dψ dβ * (0 + ) = 0, This property is in concordance with our result because the critical curve satisfied the same property, see Corollary 2.…”

“…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].…”