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

Abstract

We consider the 2-dimensional Toda lattice tau functions τ n ( t , s ; η , θ ) deforming the probabilities τ n ( 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.