Recursion Relation for Toeplitz Determinants and the Discrete Painlevé II Hierarchy

Thomas Chouteau   ; Sofia Tarricone

Symmetry, integrability and geometry: methods and applications, Tome 19 (2023)

Résumé

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.

Détail

Keywords: Riemann–Hilbert problems, Toeplitz determinants.
Mots-clés : discrete Painlevé equations, orthogonal polynomials

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