Orthogonal polynomials and discrete Painlevé equations: theory and applications

<p>In these lectures we study how discrete Painlevé equations appear in the theory of orthogonal polynomials. The occurrence of this phenomenon is due to the fact that families of orthogonal polynomials can be described via recurrence relations, either the three-term recurrence relation in the case of orthogonality measure on the real line or the Szegő recurrence relation in the case of orthogonality measure on the unit circle. As it turned out, the recurrence coefficients appearing in these relations satisfy, for certain type of (parameter-depending) measures, nonlinear discrete equations of discrete Painlevé type with coefficients depending on the parameters of the measure. In the first part of the lectures, we review the classical theory of orthogonal polynomials on the real line (OPRL for short) introducing as well the Riemann-Hilbert approach. We then focus on the particular case of orthogonality measure being a deformation of the Gaussian measure on the real line, to see how the discrete Painlevé I equation arises in this setting. In the second part of the lectures, we show how the theory of OPRL can be extended to the case of orthogonal polynomials with a measure on the unit circle in the complex plane (OPUC for short). We then focus on a specific orthogonality measure to see how this time, the discrete Painlevé II equation describes the recurrence satisfied by the Versblunsky coeffiecients of this family of orthogonal polynomials. Finally, we see some applications of these results, respectively in matrix models appearing in 2D quantum gravity and in the Ulam problem for random permutations. In particular, we see how the study of continuous limit of the discrete Painlevé I and II equations appearing in these contexts, via their relation with orthogonal polynomials, led to fundamental asymptotic results.</p>