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

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

2010-01-24

arxiv.org/abs/1001.4244v1

J. Comput. Appl. Math. 236 (2011) 19-27

Nonlinear Sciences

Exactly Solvable and Integrable Systems

15 pages

Scientific paper

10.1016/j.cam.2010.06.006

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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