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

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.

15 pages

References in corpus (2)

• Multiple Hamiltonian structure of Bogoyavlensky-Toda lattices
• The correspondence between Tracy-Widom (TW) and Adler-Shiota-van Moerbeke (ASvM) approaches in random matrix theory: the Gaussian case

Cited by in corpus (2)

• Continuous and discrete Painleve equations arising from the gap probability distribution of the finite $n$ Gaussian Unitary Ensembles
• A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy