Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 030, 30 pages      arXiv:2211.16898      https://doi.org/10.3842/SIGMA.2023.030
Contribution to the Special Issue on Evolution Equations, Exactly Solvable Models and Random Matrices in honor of Alexander Its' 70th birthday

Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy

Thomas Chouteau a and Sofia Tarricone b
a) Université d'Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France
b) Institut de Physique Théorique, Université Paris-Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, France

Received December 22, 2022, in final form May 16, 2023; Published online May 28, 2023

Abstract
Solutions of the discrete Painlevé II hierarchy are shown to be in relation with a family of Toeplitz determinants describing certain quantities in multicritical random partitions models, for which the limiting behavior has been recently considered in the literature. Our proof is based on the Riemann-Hilbert approach for the orthogonal polynomials on the unit circle related to the Toeplitz determinants of interest. This technique allows us to construct a new Lax pair for the discrete Painlevé II hierarchy that is then mapped to the one introduced by Cresswell and Joshi.

Key words: discrete Painlevé equations; orthogonal polynomials; Riemann-Hilbert problems; Toeplitz determinants.

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