Discrete Painlev\'e equations, Orthogonal Polynomials on the Unit Circle and $N$-recurrences for averages over U(N) -- \PVI $\tau$-functions

TL;DR

This paper develops a comprehensive theory connecting orthogonal polynomials on the unit circle, discrete Painlevé equations, and matrix integrals, revealing deep links between these areas through Riemann-Hilbert problems and isomonodromic deformations.

Contribution

It introduces a unified framework for orthogonal polynomials on the unit circle with semi-classical weights, deriving associated difference and differential equations, and establishing their equivalence to discrete Painlevé equations.

Findings

Difference equations are equivalent to discrete Painlevé equations for specific weights.

The Riemann-Hilbert problem formulation encapsulates the orthogonal polynomial system.

Matrix integrals over U(N) relate to hypergeometric functions and Painlevé equations.

Abstract

The theory of orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference equations of certain coefficient functions appearing in the theory. A natural formulation of the Riemann-Hilbert problem is presented which has as its solution the above system of orthogonal polynomials and associated functions. In particular for the case of regular semi-classical weights on the unit circle $ w(z) = \prod^m_{j=1}(z-z_j(t))^{\rho_j} $, consisting of $ m \in \mathbb{Z}_{> 0} $ singularities, difference equations with respect to the orthogonal polynomial degree $ n $ (Laguerre-Freud equations) and differential equations with respect to the deformation variables $ z_j(t) $ (Schlesinger equations) are derived completely characterising the system. It is shown in the simplest non-trivial case of $ m=3 $ that quite generally and simply the difference equations are equivalent to the discrete Painlev\'e equation associated with the degeneration of the rational surface $ D^{(1)}_4 \to D^{(1)}_5 $ and no other. In a three way comparison with other methods employed on this problem - the Toeplitz lattice and Virasoro constraints, the isomonodromic deformation of $ 2\times 2 $ linear Fuchsian differential equations, and the algebraic approach based upon the affine Weyl group symmetry - it is shown all are entirely equivalent, when reduced in order by exact summation, to the above discrete Painlev\'e equation through explicit transformation formulae. The fundamental matrix integrals over the unitary group $ U(N) $ arising in the theory are given by the generalised hypergeometric function $ {{}^{\vphantom{(1)}}_2}F^{(1)}_1 $.