Exact Solution of the O(n) Model on a Random Lattice

TL;DR

This paper provides an exact solution to the $O(n)$ model on a random lattice, revealing universality properties across potential functions and values of $n$, and offers explicit formulas for various correlators and free energies.

Contribution

It introduces an exact, universal solution for the $O(n)$ model on random lattices, applicable across a range of $n$ and potentials, with detailed higher-genus calculations.

Findings

Universal form of the solution for $n$ in (-2,2)

Explicit genus zero and one contributions to correlators and free energy

Method to compute higher-genus contributions and their scaling limits

Abstract

We present an exact solution of the $O(n)$ model on a random lattice. The coupling constant space of our model is parametrized in terms of a set of moment variables and the same type of universality with respect to the potential as observed for the one-matrix model is found. In addition we find a large degree of universality with respect to $n$; namely for $n\in ]-2,2[$ the solution can be presented in a form which is valid not only for any potential, but also for any $n$ (not necessarily rational). The cases $n=\pm 2$ are treated separately. We give explicit expressions for the genus zero contribution to the one- and two-loop correlators as well as for the genus one contribution to the one-loop correlator and the free energy. It is shown how one can obtain from these results any multi-loop correlator and the free energy to any genus and the structure of the higher genera contributions is described. Furthermore we describe how the calculation of the higher genera contributions can be pursued in the scaling limit.