A centerless representation of the Virasoro algebra associated with the unitary circular ensemble

Luc Haine; Didier Vanderstichelen

doi:10.1016/j.cam.2010.06.006

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arXiv · 1001.4244

A centerless representation of the Virasoro algebra associated with the unitary circular ensemble

Luc Haine·Didier Vanderstichelen

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Abstract

We consider the 2-dimensional Toda lattice tau functions $\tau_n(t,s;\eta,\theta)$ deforming the probabilities $\tau_n(\eta,\theta)$ that a randomly chosen matrix from the unitary group U(n), for the Haar measure, has no eigenvalues within an arc $(\eta,\theta)$ of the unit circle. We show that these tau functions satisfy a centerless Virasoro algebra of constraints, with a boundary part in the sense of Adler, Shiota and van Moerbeke. As an application, we obtain a new derivation of a differential equation due to Tracy and Widom, satisfied by these probabilities, linking it to the Painleve VI equation.

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Luc Haine, Didier Vanderstichelen. 2010-01-24. A centerless representation of the Virasoro algebra associated with the unitary circular ensemble. https://doi.org/10.1016/j.cam.2010.06.006

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