Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies

TL;DR

This paper computes the generating functions of the O(n) loop gas model on random lattices of all topologies, revealing they follow topological recursion and are given by symplectic invariants of their spectral curve.

Contribution

It extends the computation of generating functions for the O(n) model to all topologies, demonstrating their adherence to topological recursion.

Findings

Generating functions obey topological recursion.

Functions are given by symplectic invariants of spectral curves.

Results generalize known cases to all topologies.

Abstract

We compute the generating functions of a O(n) model (loop gas model) on a random lattice of any topology. On the disc and the cylinder, they were already known, and here we compute all the other topologies. We find that the generating functions (and the correlation functions of the lattice) obey the topological recursion, as usual in matrix models, i.e they are given by the symplectic invariants of their spectral curve.