From 1-matrix model to Kontsevich model

Abstract

Loop equations of matrix models express the invariance of the models under field redefinitions. We use loop equations to prove that it is possible to define continuum times for the generic hermitian {1-matrix} model such that all correlation functions in the double scaling limit agree with the corresponding correlation functions of the Kontsevich model expressed in terms of kdV times. In addition the double scaling limit of the partition function of the hermitian matrix model agree with the $τ$-function of the kdV hierarchy corresponding to the Kontsevich model (and not the square of the $τ$-function) except for some complications at genus zero.