Multiple orthogonal polynomials, string equations and the large-n limit

The Riemann-Hilbert problems for multiple orthogonal polynomials of types I and II are used to derive string equations associated to pairs of Lax-Orlov operators. A method for determining the quasiclassical limit of string equations in the phase space of the Whitham hierarchy of dispersionless integrable systems is provided. Applications to the analysis of the large-n limit of multiple orthogonal polynomials and their associated random matrix ensembles and models of non-intersecting Brownian motions are given.

💡 Research Summary

This paper develops a comprehensive framework for analyzing multiple orthogonal polynomials (MOPs) of type I and type II in the large‑degree (large‑n) regime by exploiting their Riemann–Hilbert (RH) formulation. The authors begin by recalling that MOPs satisfy a matrix‑valued RH problem whose jump matrices encode several weight functions simultaneously. By translating the RH problem into a pair of Lax–Orlov operators (L) and (M) (with the canonical commutation relation (