Unitary Matrix Models and Painlevé Iii

Abstract

We discussed the full unitary matrix models from the viewpoints of integrable equations and string equations. Coupling the Toda equations and the string equations, we derive a special case of the Painlevé III equation. From the Virasoro constraints, we can use the radial coordinate. The relation between t1 and t−1 is like the complex conjugate.

Discriminants and Functional Equations for Polynomials Orthogonal on the Unit Circle

“…Some recurrence relations for the corresponding coefficients of the monic version of these orthogonal polynomials have been known [26], [16], [34] and we derive the equivalent results for } n , etc.…”

Asymptotic behavior of the Verblunsky coefficients for the OPUC with a varying weight

“…The result was obtained in [5], but in some different form, therefore we give its proof in Section 3. We use (1.13) as a system of nonlinear equations for α (n) k and solve it by the perturbation theory method.…”

Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials

“…Periwal and Shevitz [20] found a non-linear recurrence relation for the Verblunsky coefficients of these orthogonal polynomials (see also [9,23]). If z = e iθ then…”

“…It was Magnus [17] who made the connection with the discrete Painlevé equation I. The recurrence relation in Section 3 was found by Periwal and Shevitz [20] (see also Hisakado [9], Tracy and Widom [23]; Baik [1] used the Riemann-Hilbert approach to obtain the Painlevé equation). The recurrence relations in Section 4 were obtained by Van Assche and Foupouagnigni in [25].…”