Loop models on random maps via nested loops: case of domain symmetry breaking and application to the Potts model

TL;DR

This paper investigates loop models on random maps with domain symmetry breaking, deriving functional relations, analyzing phase diagrams, and applying findings to the Potts model, revealing non-self-duality and spontaneous symmetry breaking phenomena.

Contribution

It introduces a nested loop approach to study domain symmetry breaking in loop models on random maps, extending complex analytic techniques to solve coupled functional relations.

Findings

Derived coupled functional relations for loop configurations on maps.

Analyzed phase diagram and identified non-generic critical points.

Showed the Potts model on random maps is not self-dual for Q ≠ 1.

Abstract

We use the nested loop approach to investigate loop models on random planar maps where the domains delimited by the loops are given two alternating colors, which can be assigned different local weights, hence allowing for an explicit Z_2 domain symmetry breaking. Each loop receives a non local weight n, as well as a local bending energy which controls loop turns. By a standard cluster construction that we review, the Q = n^2 Potts model on general random maps is mapped to a particular instance of this problem with domain-non-symmetric weights. We derive in full generality a set of coupled functional relations for a pair of generating series which encode the enumeration of loop configurations on maps with a boundary of a given color, and solve it by extending well-known complex analytic techniques. In the case where loops are fully-packed, we analyze in details the phase diagram of the model and derive exact equations for the position of its non-generic critical points. In particular, we underline that the critical Potts model on general random maps is not self-dual whenever Q \neq 1. In a model with domain-symmetric weights, we also show the possibility of a spontaneous domain symmetry breaking driven by the bending energy.