Critical behaviour of loop models on causal triangulations

TL;DR

This paper introduces dense and dilute loop models on causal triangulations, maps them to planar tree models, and analyzes their critical behavior using explicit calculations and transfer matrix methods.

Contribution

It presents the first mapping of loop models on causal triangulations to planar tree models and analyzes their critical points, especially for the dilute case.

Findings

Dense loop model mapped to a solvable planar tree model with explicit partition function.

Bounds on critical coupling $g_c$ derived for the dilute loop model.

Critical behavior examined for small $eta$ using transfer matrix techniques.

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.