From 1-matrix model to Kontsevich model

TL;DR

This paper demonstrates that the continuum limit of the hermitian 1-matrix model aligns with the Kontsevich model's correlation functions and tau-function, establishing a deep connection via loop equations and double scaling limits.

Contribution

It proves the equivalence of the hermitian 1-matrix model's double scaling limit with the Kontsevich model's correlation functions and tau-function, using loop equations.

Findings

Correlation functions match in the double scaling limit

Partition functions agree with the Kontsevich tau-function

Results hold except for some genus zero complications

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 $\tau$-function of the kdV hierarchy corresponding to the Kontsevich model (and not the square of the $\tau$-function) except for some complications at genus zero.