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Orthogonal Polynomials in the Normal Matrix Model with a Cubic Potential

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Orthogonal Polynomials in the Normal Matrix Model with a Cubic Potential CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Available online at www.sciencedirect.com Advances in Mathematics 230 (2012) 1272–1321 www.elsevier.com/locate/aim Orthogonal polynomials in the normal matrix model with a cubic potential Pavel M. Blehera, Arno B.J. Kuijlaarsb,∗ a Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, IN 46202, USA b Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B bus 2400, 3001 Leuven, Belgium Received 29 June 2011; accepted 6 March 2012 Available online 23 April 2012 Communicated by Nikolai Makarov Abstract 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 n 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 (µ1; µ2/ on the complex plane. We then formulate a 3 × 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 (µ1; µ2/, and the construction of a global parametrix. The main result of this paper is a derivation of the large n 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 µ1, the first component of the equilibrium measure. We also obtain analytical results forthe ∗ Corresponding author. E-mail addresses: [email protected] (P.M. Bleher), [email protected] (A.B.J. Kuijlaars). 0001-8708/$ - see front matter ⃝c 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2012.03.021 P.M. Bleher, A.B.J. Kuijlaars / Advances in Mathematics 230 (2012) 1272–1321 1273 measure µ1 relating it to the distribution of eigenvalues in the normal matrix model which is uniform in a domain bounded by a simple closed curve. ⃝c 2012 Elsevier Inc. All rights reserved. Keywords: Normal matrix model; Orthogonal polynomials; Vector equilibrium problem; Riemann–Hilbert problem; Steepest descent analysis 1. Introduction The normal matrix model is a probability measure on the space of n × n normal matrices M of the form 1 exp .−nTrV.M// d M: (1.1) Zn A typical form for V is 1 ∗ ∗ V.M/ D MM − V .M/ − V .M / ; t0 > 0; (1.2) t0 where V is a polynomial and V is the polynomial obtained from V by conjugating the coefficients. In this case, the model may alternatively be defined ongeneral n × n complex matrices M, see e.g. [34]. In this paper we study in particular (1.1) and (1.2) with a cubic potential t V .M/ D 3 M3; t > 0: (1.3) 3 3 The main feature of the normal matrix model is that the eigenvalues of M fill out a bounded two-dimensional domain Ω as n ! 1 with a uniform density. Wiegmann and Zabrodin [32] showed that if 1 X tk V .M/ D Mk (1.4) k kD1 then Ω D Ω.t0I t1; t2; : : :/ is such that 1 1 ZZ d A.z/ D D − D t0 area.Ω/; tk k ; k 1; 2; 3;::: (1.5) π π CnΩ z where d A denotes the two-dimensional Lebesgue measure in the complex plane. The relations (1.5) characterize the domain Ω by means of its area πt0 and its exterior harmonic moments tk for k ≥ 1 (it is assumed that 0 2 Ω, and the integrals in (1.5) need to be regularized for k ≤ 2). An important fact, first shown in [22], is that the boundary of Ω as a function of t0 > 0 (which is seen as a time parameter) evolves according to the model of Laplacian growth. The Laplacian growth is unstable and singularities such as boundary cusps may appear in finite time. See also [27,30,33,34] for related work and surveys. However, from a mathematical point of view, the model (1.1) and (1.2) is not well-defined if V is a polynomial of degree ≥ 3, since then the integral Z Zn D exp.−nTrV.M//d M (1.6) 1274 P.M. Bleher, A.B.J. Kuijlaars / Advances in Mathematics 230 (2012) 1272–1321 diverges, and one cannot normalize the measure (1.1) to make it a probability measure. To make the model well-defined, Elbau and Felder [17,18] propose to use a cut-off procedure. Instead of considering all normal matrices M, they restrict to normal matrices with spectrum in a fixed bounded domain D. The integral (1.6) restricted to all such matrices is convergent and the model is well-defined. As it is the case for unitary random matrices, the eigenvalues of M are then distributed according to a determinantal point process with a kernel that is built out of orthogonal polynomials with respect to the scalar product ZZ −nV.z/ h f; giD D f .z/g.z/e d A.z/ (1.7) D (which depends on n), with 1 V.z/ D jzj2 − V .z/ − V .z/ ; (1.8) t0 1 D see [17]. For each n, we have the sequence .Pk;n/kD0 of monic polynomials (i.e., Pk;n.z/ zk C···) such that hPk;n; Pj;niD D hk;nδ j;k; and then the correlation kernel for the determinantal point process is n−1 − n .V.w/CV.z// X Pk;n.z/Pk;n.w/ K .w; z/ D e 2 : n h kD0 k;n Elbau and Felder [17,18] prove that for a polynomial V as in (1.4) with t1 D 0, jt2j < 1, and for t0 small enough it is possible to find a suitable domain D such that indeed the eigenvalues of the normal matrix model with cut-off D accumulate on a domain Ω as n ! 1. The domain Ω is characterized by (1.5) and so in particular evolves according to Laplacian growth in the time parameter t0. Note that the cut-off approach works fine for t0 small enough but fails to capture important features of the normal matrix model such as the formation of cusp singularities at a critical value of t0. Elbau [17] also discusses the zeros of the orthogonal polynomials Pn;n as n ! 1. For the cubic potential (1.3) and again for t0 sufficiently small, he shows that these zeros accumulate on a starlike set ∗ ∗ 2 ∗ 2πi=3 Σ1 DT0; x U[T0;!x U[T0;! x U;! D e ; (1.9) ∗ for some explicit value of x > 0. The set Σ1 is contained in Ω. The limiting distribution of ∗ zeros is a probability measure µ1 on Σ1 satisfying Z ZZ ∗ 1 log jz − ζj dµ1(ζ / D log jz − ζj d A(ζ /; z 2 C n Ω: (1.10) πt0 Ω For general polynomial V and t0 small enough, Elbau conjectures that the zeros of the orthogonal polynomials accumulate on a tree-like set strictly contained in Ω with a limiting distribution whose logarithmic potentials outside of Ω agrees with that of the normalized Lebesgue measure on Ω. P.M. Bleher, A.B.J. Kuijlaars / Advances in Mathematics 230 (2012) 1272–1321 1275 In this paper we want to analyze the orthogonal polynomials for the normal matrix model without making a cut-off. We cannot use the scalar product ZZ h f; gi D f .z/g.z/e−nV.z/d A.z/ (1.11) C defined on C since the integral (1.11) diverges if f and g are polynomials and V is a polynomial of degree ≥ 3. Our approach is to replace (1.11) by a Hermitian form defined on polynomials that is a priori not given by any integral, but that should satisfy the relevant algebraic properties of the scalar product (1.11). We define and classify these Hermitian forms. See the next section for precise statements. It turns out that there is more than one possibility for such a Hermitian form. We conjecture that for any polynomial V it is possible to choose the Hermitian form in such a way that the corresponding orthogonal polynomials have the same asymptotic behavior as the orthogonal polynomials in the cut-off approach of Elbau and Felder. That is, for small values of t0 the zeros ∗ of the polynomials accumulate on a tree like set Σ1 with a limiting distribution µ1 satisfying (1.10). We are able to establish this for the cubic case (1.3) and this is the main result of the paper. We recover the same set Σ1 as in (1.9) and also the domain Ω that evolves according to the Laplacian growth, as we will show.
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