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

Haine, Luc;Vanderstichelen, Didier
(2011) Journal of Computational and Applied Mathematics —

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  • Haine, LucUCLouvain
    Author
  • Vanderstichelen, DidierUCLouvain
    Author
Abstract
We consider the 2-dimensional Toda lattice tau functions τn(t,s;η,θ) deforming the probabilities τn(η,θ) that a randomly chosen matrix from the unitary group U(n), for the Haar measure, has no eigenvalues within an arc (η,θ) 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 Painlevé VI equation.
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Haine, L., & Vanderstichelen, D. (2011). A centerless representation of the Virasoro algebra associated with the unitary circular ensemble. Journal of Computational and Applied Mathematics. https://doi.org/10.1016/j.cam.2010.06.006