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

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

2007-12-06

arxiv.org/abs/0712.0862v1

Journal of Mathematical Physics, 49 (2008), 043503

Physics

Mathematical Physics

20 pages, submitted to Journal of Mathematical Physics

Scientific paper

10.1063/1.2890428

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.

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