The correspondence between Tracy-Widom (TW) and Adler-Shiota-van Moerbeke (ASvM) approaches in random matrix theory: the Gaussian case

TL;DR

This paper compares two methods, TW and ASvM, for deriving integrable differential equations in Gaussian random matrix theory, revealing explicit relations and core structures like orthogonal functions and Toda lattice.

Contribution

It establishes explicit connections between TW and ASvM approaches, clarifying their relationship through orthogonal functions and Toda lattice in Gaussian ensembles.

Findings

Explicit relations between TW variables and ASvM τ-functions

Identification of orthogonal functions as core structure

Unified view of integrable equations in Gaussian matrices

Abstract

Two approaches (TW and ASvM) to derivation of integrable differential equations for random matrix probabilities are compared. Both methods are rewritten in such a form that simple and explicit relations between all TW dependent variables and $\tau$-functions of ASvM are found, for the example of finite size Gaussian matrices. Orthogonal function systems and Toda lattice are seen as the core structure of both approaches and their relationship.