THE MULTICRITICAL KONTSEVICH-PENNER MODEL

Abstract

We consider the Hermitian matrix model with an external field entering the quadratic term tr (ΛXΛX) and Penner-like interaction term αN( log (1+X)-X). An explicit solution in the leading order in N is presented. The critical behavior is given by the second derivative of the free energy in α which appears to be a pure logarithm, that is a feature of c=1 theories. Various critical regimes are possible, some of them corresponds to critical points of the usual Penner model, but there exists an infinite set of multicritical points which differ by values of scaling dimensions of proper conformal operators. Their correlators with the puncture operator are given in genus zero by Legendre polynomials whose argument is determined by an analog of the string equation.