Critical behaviour of loop models on causal triangulations

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 $\alpha$ in such a way that the purely geometric causal triangulation model is recovered for $\alpha =1$. 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 $\alpha$ small.

30 pages, v2: minor changes

References in corpus (4)

• Shaken, but not stirred - Potts model coupled to quantum gravity
• Combinatorial point for higher spin loop models
• A restricted dimer model on a 2-dimensional random causal triangulation
• Automorphism Induced Nonlocal Conservation Laws

Cited by in corpus (3)

• Trees with exponential height dependent weight
• The cylinder amplitude in the Hard Dimer model on 2D Causal Dynamical Triangulations
• Trees with power-like height dependent weight