Diffusion Equation, Continuum Limit and Universality in Two Dimensional Quantum Gravity

Abstract

In this paper we will present a brief summary of the definition of the continuum limit of the matrix model description of two dimensional gravity using a diffusion equation. This method differs from the continuum limit originally presented in [1, 2, 7] in the following way: the continuum limit in these papers is obtained by firstly computing the matrix elements of a certain operator and subsequently taking the continuum limit of the matrix elements. In this article we use a different approach. We derive an exact difference equation satisfied by the matrix elements and subsequently take the continuum limit of this equation. The continuum differential equation is then used to define the continuum limit of the original matrix elements in a totally consistent way. Both these methods lead to the same universal behavior as seen by explicit examples.

This work was supported in part by the Centre National de la Recherche Scientifique.