Discrete Painlev\'e equations for a class of \PVI $\tau$-functions given as U(N) averages

TL;DR

This paper derives discrete Painlevé equations for U(N) averages related to bi-orthogonal polynomials on the unit circle, connecting difference equations with integrable systems and applications in random matrix theory and statistical physics.

Contribution

It shows that for three singularities, the difference equations reduce to a coupled system that can be transformed into the discrete fifth Painlevé equation, linking various mathematical frameworks.

Findings

Reduction to discrete fifth Painlevé equation for specific cases

Explicit mapping between different forms of PV

Applications to random matrix gap probabilities and Ising model correlations

Abstract

In a recent work difference equations (Laguerre-Freud equations) for the bi-orthogonal polynomials and related quantities corresponding to the weight on the unit circle $ w(z)=\prod^m_{j=1}(z-z_j(t))^{\rho_j} $ were derived.Here it is shown that in the case $ m=3 $ these difference equations, when applied to the calculation of the underlying U(N) average, reduce to a coupled system identifiable with that obtained by Adler and van Moerbeke using methods of the Toeplitz lattice and Virasoro constraints. Moreover it is shown that this coupled system can be reduced to yield the discrete fifth Painlev\'e equation \dPV as it occurs in the theory of the sixth Painlev\'e system. Methods based on affine Weyl group symmetries of B\"acklund transformations have previously yielded the \dPV equation but with different parameters for the same problem. We find the explicit mapping between the two forms. Applications of our results are made to give recurrences for the gap probabilities and moments in the circular unitary ensemble of random matrices, and to the diagonal spin-spin correlation function of the square lattice Ising model.