Sums of random matrices and the Potts model on random planar maps

Abstract

We compute the partition function of the q-states Potts model on a random planar lattice with 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 and comment on the conformal field theory description of the critical points.