More on the exact solution of the O(n) model on a random lattice and an investigation of the case |n|>2

Abstract

For $n\in [-2,2]$ the $O(n)$ model on a random lattice has critical points to which a scaling behaviour characteristic of 2D gravity interacting with conformal matter fields with $c\in [-\infty,1]$ can be associated. Previously we have written down an exact solution of this model valid at any point in the coupling constant space and for any $n$. The solution was parametrized in terms of an auxiliary function. Here we determine the auxiliary function explicitly as a combination of $θ$-functions, thereby completing the solution of the model. Using our solution we investigate, for the simplest version of the model, hitherto unexplored regions of the parameter space. For example we determine in a closed form the eigenvalue density without any assumption of being close to or at a critical point. This gives a generalization of the Wigner semi-circle law to $n\neq 0$. We also study the model for $|n|>2$. Both for $n<-2$ and $n>2$ we find that the model is well defined in a certain region of the coupling constant space. For $n<-2$ we find no new critical points while for $n>2$ we find new critical points at which the string susceptibility exponent $γ_{str}$ takes the value $+\frac{1}{2}$.