Boundary Conditions and the q-state Potts model on Random Planar Maps

TL;DR

This paper analyzes the boundary conditions and critical behavior of the q-state Potts model on random planar maps, revealing new solutions and dualities, and connecting them to known boundary conditions on flat lattices.

Contribution

It extends the analysis of the Potts model on random maps to new boundary conditions and dualities, identifying novel solutions and their relation to flat lattice boundary conditions.

Findings

Identifies allowed q-values as q=2(1+cos(m/n)π) with coprime m,n.

Discovers two sequences of solutions with different boundary conditions.

Establishes duality relations between boundary conditions, including the 'New' boundary condition.

Abstract

We extend a recent analysis of the $q$-states Potts model on an ensemble of random planar graphs with $p\leqslant q$ allowed, equally weighted, spins on a connected boundary. In this paper we explore the $(q<4,p\leqslant q)$ parameter space of finite-sheeted resolvents and derive the associated critical exponents. By definition a value of $q$ is allowed if there is a $p=1$ solution, and we reproduce the long-known result that $q= 2(1+\cos{\frac{m}{n} \pi})$ with $m,n$ coprime. In addition we find that there are two distinct sequences of solutions, one of which contains $p=2$ and $p=q/2$ while the other does not. The boundary condition $p=3$ appears only for $q=3$ which also has a $p=3/2$ boundary condition; we conjecture that this new solution corresponds in the scaling limit to the 'New' boundary condition, discovered on the flat lattice by Affleck et al. We also explore Kramers-Wannier duality for $q=3$ in this context and explicitly construct the known boundary conditions; we show that the mixed boundary condition is dual to a boundary condition on dual graphs that corresponds to Affleck et al's identification of the New boundary condition on fixed lattices. On the other hand we find that the mixed boundary condition of the dual, and the corresponding New boundary condition of the original theory are not described by conventional resolvents.