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.

arXiv comment30 pages, v2: minor changes

References in corpus (28)

• Non-perturbative Lorentzian Quantum Gravity, Causality and Topology Change
• Logarithmic Minimal Models
• Planar algebras, I
• Scaling in quantum gravity
• Exact Solution of the O(n) Model on a Random Lattice
• A second-order phase transition in CDT
• Multicritical Phases of the O(n) Model on a Random Lattice
• Solvable Critical Dense Polymers
• A new perspective on matter coupling in 2d quantum gravity
• The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds
• Integrable 2D Lorentzian Gravity and Random Walks
• On the spectral dimension of causal triangulations
• Shaken, but not stirred - Potts model coupled to quantum gravity
• Q-states Potts model on a random planar lattice
• Exact S-matrices for supersymmetric sigma models and the Potts model
• Solvable Critical Dense Polymers on the Cylinder
• Combinatorial point for higher spin loop models
• Loop models on random maps via nested loops: case of domain symmetry breaking and application to the Potts model
• An Analytical Analysis of CDT Coupled to Dimer-like Matter
• Logarithmic Superconformal Minimal Models
• Fusion hierarchies, T-systems and Y-systems of logarithmic minimal models
• Modular invariant partition function of critical dense polymers
• Fully Packed O(n=1) Model on Random Eulerian Triangulations
• Upper bounds on Liouville first passage percolation and Watabiki's prediction
• Bounds on the critical line via transfer matrix methods for an Ising model coupled to causal dynamical triangulations
• A restricted dimer model on a 2-dimensional random causal triangulation
• Generic Ising Trees
• 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