Matrix Models of Two-dimensional Gravity and Discrete Toda Theory

TL;DR

This paper establishes a connection between one-matrix models of two-dimensional gravity and discrete Toda theory, showing that their recursion relations correspond to Toda hierarchy equations with Virasoro constraints, without requiring the double scaling limit.

Contribution

It demonstrates that the recursion relations in matrix models are equivalent to discrete Toda hierarchy equations with Virasoro constraints, introducing a discrete time variable and tau functions.

Findings

Recursion relations match Toda-chain hierarchy equations.

Partition functions are tau functions satisfying discrete Toda molecule equations.

Relations between matrix models and discrete Toda theory are clarified.

Abstract

Recursion relations for orthogonal polynominals, arising in the study of the one-matrix model of two-dimensional gravity, are shown to be equvalent to the equations of the Toda-chain hierarchy supplemented by additional Virasoro constraints. This is the case without the double scaling limit. A discrete time variable to the matrix model is introduced. The discrete time dependent partition functions are given by $\tau$ functions which satisfy the discrete Toda molecule equation. Further the relations between the matrix model and the discrete time Toda theory are discussed.