Critical behaviour of loop models on causal triangulations
Bergfinnur Durhuus, Xavier Poncini, Jørgen Rasmussen, Meltem Ünel · Journal of Statistical Mechanics Theory and Experiment · 2021
Abstract We introduce a dense and a dilute loop model on causal dynamical triangulations. Both models are characterised by a geometric coupling constant g and a loop parameter α in such a way that the purely geometric causal triangulation model is recovered for α = 1. We show that the dense loop model can be mapped to a solvable planar tree model, whose partition function we compute explicitly and use to determine the critical behaviour of the loop model. The dilute loop model can likewise be mapped to a planar tree model; however, a closed-form expression for the corresponding partition function is not obtainable using the standard methods employed in the dense case. Instead, we derive bounds on the critical coupling g c and apply transfer matrix techniques to examine the critical behaviour for α small.