The $O(n)$ model on a random surface: critical points and large order behaviour

TL;DR

This paper investigates the critical behavior and large order expansion of an $O(n)$ symmetric matrix model on a random surface, classifying critical points and deriving string susceptibility exponents in the large $N$ limit.

Contribution

It extends the analysis of $O(n)$ models on random lattices by classifying critical points and calculating string susceptibility exponents using steepest descent and double scaling techniques.

Findings

Critical points classified by two integers p,q

String susceptibility exponent formula derived

Results match known $(p,q)$ string models when l=1

Abstract

In this article we report a preliminary investigation of the large $N$ limit of a generalized one-matrix model which represents an $O(n)$ symmetric model on a random lattice. The model on a regular lattice is known to be critical only for $-2\le n\le 2$. This is the situation we shall discuss also here, using steepest descent. We first determine the critical and multicritical points, recovering in particular results previously obtained by Kostov. We then calculate the scaling behaviour in the critical region when the cosmological constant is close to its critical value. Like for the multi-matrix models, all critical points can be classified in terms of two relatively prime integers $p,q$. In the parametrization $p=(2m+1)q \pm l$, $m,l$ integers such that $0<l<q$, the string susceptibility exponent is found to be $\gamma_{\rm string}=-2l/(p+q-l)$. When $l=1$ we find that all results agree with those of the corresponding $(p,q)$ string models, otherwise they are different.\par We finally explain how to derive the large order behaviour of the corresponding topological expansion in the double scaling limit.