Recursion Relation for Toeplitz Determinants and the Discrete Painlev\'e II Hierarchy

TL;DR

This paper establishes a connection between solutions of the discrete Painlevé II hierarchy and Toeplitz determinants, using Riemann-Hilbert techniques to construct a new Lax pair and relate it to existing formulations.

Contribution

It introduces a novel Riemann-Hilbert approach to link Toeplitz determinants with the discrete Painlevé II hierarchy and constructs a new Lax pair for this integrable system.

Findings

Linked Toeplitz determinants to discrete Painlevé II solutions

Developed a new Lax pair for the hierarchy

Mapped the new Lax pair to existing formulations

Abstract

Solutions of the discrete Painlev\'e 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 Painlev\'e II hierarchy that is then mapped to the one introduced by Cresswell and Joshi.