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

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.

20 pages, submitted to Journal of Mathematical Physics

References in corpus (2)

• Differential Equations for Dyson Processes
• PDEs for the joint distributions of the Dyson, Airy and Sine processes

Cited by in corpus (3)

• A centerless representation of the Virasoro algebra associated with the unitary circular ensemble
• Universal Structure and Universal PDE for Unitary Ensembles
• Various equations for gap probabilities of coupled Gaussian matrices