Boundary States of the Potts Model on Random Planar Maps

TL;DR

This paper analyzes the 3-states Potts model on random planar maps using matrix models, deriving an algebraic curve for boundary conditions, studying critical behavior, and connecting it to Liouville quantum gravity and minimal models.

Contribution

It introduces an algebraic curve encoding boundary conditions for the Potts model on random maps and explores its critical behavior and conformal field theory description.

Findings

Derived algebraic curve for boundary conditions.

Identified critical exponents consistent with previous studies.

Linked the double scaling limit to Liouville quantum gravity with extended symmetry.

Abstract

We revisit the 3-states Potts model on random planar triangulations as a Hermitian matrix model. As a novelty, we obtain an algebraic curve which encodes the partition function on the disc with both fixed and mixed spin boundary conditions. We investigate the critical behaviour of this model and find scaling exponents consistent with previous literature. We argue that the conformal field theory that describes the double scaling limit is Liouville quantum gravity coupled to the $(A_4,D_4)$ minimal model with extended $\mathcal{W}_3$-symmetry.