Orthogonal polynomials in the normal matrix model with a cubic potential

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• 作者：Pavel M. Bleher^a ; ^{bleher@math.iupui.edu} ; Arno B.J. Kuijlaars^b ; ^{arno.kuijlaars@wis.kuleuven.be}
• 关键词：Normal matrix model ; Orthogonal polynomials ; Vector equilibrium problem ; Riemann&ndash ; Hilbert problem ; Steepest descent analysis
• 刊名：Advances in Mathematics
• 出版年：2012
• 期刊代码：62_00018708
• 类别：mt
• 出版时间：20 June, 2012
• 卷：230
• 期：3
• 页码：1272-1321
• 文件大小：480 K

摘要

We consider the normal matrix model with a cubic potential. The model is ill-defined, and in order to regularize it, Elbau and Felder introduced a model with a cut-off and corresponding system of orthogonal polynomials with respect to a varying exponential weight on the cut-off region on the complex plane. In the present paper we show how to define orthogonal polynomials on a specially chosen system of infinite contours on the complex plane, without any cut-off, which satisfy the same recurrence algebraic identity that is asymptotically valid for the orthogonal polynomials of Elbau and Felder.

The main goal of this paper is to develop the Riemann-Hilbert (RH) approach to the orthogonal polynomials under consideration and to obtain their asymptotic behavior on the complex plane as the degree of the polynomial goes to infinity. As the first step in the RH approach, we introduce an auxiliary vector equilibrium problem for a pair of measures on the complex plane. We then formulate a 3脳3 matrix valued RH problem for the orthogonal polynomials in hand, and we apply the nonlinear steepest descent method of Deift-Zhou to the asymptotic analysis of the RH problem. The central steps in our study are a sequence of transformations of the RH problem, based on the equilibrium vector measure , and the construction of a global parametrix.

The main result of this paper is a derivation of the large asymptotics of the orthogonal polynomials on the whole complex plane. We prove that the distribution of zeros of the orthogonal polynomials converges to the measure , the first component of the equilibrium measure. We also obtain analytical results for the measure relating it to the distribution of eigenvalues in the normal matrix model which is uniform in a domain bounded by a simple closed curve.