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

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.