Critical region for an Ising model coupled to causal triangulations

This paper extends the results obtained by Hernández et al for the annealed Ising model coupled to two-dimensional causal dynamical triangulations. We employ the Fortuin-Kasteleyn (FK) representation in order to determine a region in the quadrant of the parameters where the critical curve for the annealed model is possibly located. This can be done by outlining a region where the model has a unique infinite-volume Gibbs measure, and a region where the finite-volume Gibbs measure does not have weak limit (in fact, does not exist if the volume is large enough). We also improve the region where the model has a one dimensional geometry with respect to the unique weak limit measure, which implies that the Ising model on causal triangulation does not have phase transition in this region. Furthermore, we provide a better approximation of the free energy for the coupled model.