A Numerical Test of KPZ Scaling: Potts Models Coupled to Two-Dimensional Quantum Gravity

TL;DR

This study uses Monte Carlo simulations to test KPZ scaling in Potts models coupled to 2D quantum gravity, confirming theoretical predictions and revealing phase transition behaviors.

Contribution

It provides numerical evidence supporting KPZ scaling for various Potts models on dynamical graphs, including cases without exact solutions, and analyzes their geometric properties.

Findings

Critical exponents agree with exact and KPZ predictions.

q=10 Potts model shows first order transition on dynamical graphs.

Graph geometry correlates with the Potts model's central charge.

Abstract

We perform Monte Carlo simulations using the Wolff cluster algorithm of the q=2 (Ising), 3, 4 and q=10 Potts models on dynamical phi-cubed graphs of spherical topology with up to 5000 nodes. We find that the measured critical exponents are in reasonable agreement with those from the exact solution of the Ising model and with those calculated from KPZ scaling for q=3,4 where no exact solution is available. Using Binder's cumulant we find that the q=10 Potts model displays a first order phase transition on a dynamical graph, as it does on a fixed lattice. We also examine the internal geometry of the graphs generated in the simulation, finding a linear relationship between ring length probabilities and the central charge of the Potts model