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The Jacobi polynomial ensemble and the Painleve VI equation

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Haine, Luc [UCL] Semengue, JP

We consider the Jacobi polynomial ensemble of n X n random matrices. We show that the probability of finding no eigenvalues in the interval [-1,z] for a random matrix chosen from the ensemble, viewed as a function of z, satisfies a second-order differential equation. After a simple change of variable, this equation can be reduced to the Okamoto-Jimbo-Miwa form of the Painleve VI equation. The result is achieved by a comparison of the Tracy-Widom and the Virasoro approaches to the problem, which both lead to different third-order differential equations. The Virasoro constraints satisfied by the tau functions are obtained by a systematic use of the moments, which drastically simplifies the computations. (C) 1999 American Institute of Physics. [S0022-2488(99)02203-3].

Document type Article de périodique (Journal article) – Article de recherche
Publication date 1999
Language Anglais
Journal information "Journal of Mathematical Physics" - Vol. 40, no. 4, p. 2117-2134 (1999)
Peer reviewed yes
Publisher Amer Inst Physics (Woodbury)
issn 0022-2488
e-issn 1089-7658
Publication status Publié
Affiliation UCL
Links
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Bibliographic reference Haine, Luc ; Semengue, JP. The Jacobi polynomial ensemble and the Painleve VI equation. In: Journal of Mathematical Physics, Vol. 40, no. 4, p. 2117-2134 (1999)
Permanent URL http://hdl.handle.net/2078.1/44542