Geometric and Stochastic Clusters of Gravitating Potts Models

TL;DR

This paper investigates the fractal dimensions of critical clusters in q-state Potts models coupled with 2D quantum gravity, combining theoretical predictions with numerical simulations, especially for the Ising case.

Contribution

It extends the KPZ formalism to Potts models on random graphs and confirms predictions through numerical simulations for the q=2 case.

Findings

Fractal dimensions of geometric clusters are related to tricritical Potts models.

Numerical simulations confirm theoretical predictions for the Ising case.

Coupling to quantum gravity alters cluster properties compared to regular lattices.

Abstract

We consider the fractal dimensions of critical clusters occurring in configurations of a q-state Potts model coupled to the planar random graphs of the dynamical triangulations formulation of Euclidean quantum gravity in two dimensions. For regular lattices, it is well-established that at criticality the properties of Fortuin-Kasteleyn clusters are directly related to the conventional critical exponents, whereas the corresponding properties of the geometric clusters of like spins are not. Recently it has been observed that the latter are related to the critical properties of a tricritical Potts model with the same central charge. We apply the KPZ formalism to develop a related prediction for the case of Potts models coupled to quantum gravity and employ numerical simulation methods to confirm it for the Ising case q=2.