On the Yang-Lee and Langer singularities in the O(n) loop model

TL;DR

This paper investigates the analytic structure of the O(n) loop model's free energy in complex magnetic fields, revealing Yang-Lee edges and Langer branch cuts, and confirming conjectures about metastability and singularities.

Contribution

It introduces a coupling to 2D quantum gravity approach to analyze the O(n) loop model, providing new insights into its singularities and metastable states.

Findings

Identification of Yang-Lee edges on the high-temperature sheet

Discovery of a Langer type branch cut on the low-temperature sheet

Confirmation of conjectures relating singularities to metastability

Abstract

We use the method of `coupling to 2d QG' to study the analytic properties of the universal specific free energy of the O(n) loop model in complex magnetic field. We compute the specific free energy on a dynamical lattice using the correspondence with a matrix model. The free energy has a pair of Yang-Lee edges on the high-temperature sheet and a Langer type branch cut on the low-temperature sheet. Our result confirms a conjecture by A. and Al. Zamolodchikov about the decay rate of the metastable vacuum in presence of Liouville gravity and gives strong evidence about the existence of a weakly metastable state and a Langer branch cut in the O(n) loop model on a flat lattice. Our results are compatible with the Fonseca-Zamolodchikov conjecture that the Yang-Lee edge appears as the nearest singularity under the Langer cut.