The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

Abstract

We investigate the asymptotic behavior of a family of multiple orthogonal polynomials that is naturally linked with the normal matrix model with a monomial potential of arbitrary degree $d+1$. The polynomials that we investigate are multiple orthogonal with respect to a system of $d$ analytic weights defined on a symmetric $(d+1)$-star centered at the origin. In the first part we analyze in detail a vector equilibrium problem involving a system of $d$ interacting measures $({\mu }_1,\dots ,{\mu }_d)$ supported on star-like sets in the plane. We show that in the subcritical regime, the first component ${\mu }_1^{\ast }$ of the solution to this problem is the asymptotic zero distribution of the multiple orthogonal polynomials. It also characterizes the domain where the eigenvalues in the normal matrix model accumulate, in the sense that the Schwarz function associated with the boundary of this domain can be expressed explicitly in terms of ${\mu }_1^{\ast }$. The second part of the paper is devoted to the asymptotic analysis of the multiple orthogonal polynomials. The asymptotic results are obtained again in the subcritical regime, and they follow from the Deift/Zhou steepest descent analysis of a Riemann-Hilbert problem of size $(d+1)\times (d+1)$. The vector equilibrium problem and the Riemann-Hilbert problem that we investigate are generalizations of those studied recently by Bleher-Kuijlaars in the case $d=2$.

65 pages, 10 figures, change in title, proof of Lemma 8.2 improved

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• Normal random matrix ensemble as a growth problem
• Random Matrices in 2D, Laplacian Growth and Operator Theory
• Viscous shocks in Hele-Shaw flow and Stokes phenomena of the Painleve I transcendent
• Shocks and finite-time singularities in Hele-Shaw flow
• Normal matrix models, dbar-problem, and orthogonal polynomials on the complex plane

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