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 $p\le q$ 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 $0\le q\le 4$ and comment on the conformal field theory description of the critical points.

31 pages, 4 figures, typos corrected