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

TL;DR

This paper links the probabilities of eigenvalue gaps in random unitary matrices to a centerless Virasoro algebra, deriving a differential equation related to Painleve VI, thus connecting random matrix theory with integrable systems.

Contribution

It introduces a novel centerless Virasoro algebra framework for these probabilities and provides a new derivation of the associated differential equation.

Findings

Tau functions satisfy a centerless Virasoro algebra of constraints.

Derived a differential equation linked to Painleve VI.

Connected eigenvalue gap probabilities to integrable systems.

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.