Sums of Random Matrices and the Potts Model on Random Planar Maps

TL;DR

This paper calculates the partition function of the Potts model on random planar maps using matrix models, revealing algebraic relations and analyzing phase transitions for 0 to 4 states, with implications for conformal field theory.

Contribution

It extends matrix model techniques to the Potts model on random maps, generalizing Voiculescu's addition of random matrices beyond free probability, and explores the model's phase diagram.

Findings

Partition functions with p and q-p colors are algebraically related.

Phase diagram analyzed for 0 ≤ q ≤ 4.

Comments on conformal field theory at critical points.

Abstract

We compute the partition function of the $q$-states Potts model on a random planar lattice with $p\leq q$ allowed, equally weighted colours on a connected boundary. To this end, we employ its matrix model representation in the planar limit, generalising a result by Voiculescu for the addition of random matrices to a situation beyond free probability theory. We show that the partition functions with $p$ and $q-p$ colours on the boundary are related algebraically. Finally, we investigate the phase diagram of the model when $0\leq q\leq 4$ and comment on the conformal field theory description of the critical points.