The Szegő curve and Laguerre polynomials with large negative parameters

Mendoza, Carlos Díaz; Orive, R.

doi:10.1016/j.jmaa.2010.12.053

Cited by 11 publications

(14 citation statements)

References 22 publications

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“…However, the case t = 1 can be independently studied in terms of the asymptotic zero distribution of Laguerre polynomials in the limit (48) with A = −1. This zero distribution was determined in [28] and also exhibits a fine structure (see Theorem 1 of [28]). Indeed, the support of the zero density ρ(z) depends on the sequence α n , but now through the parameter m = lim…”

Section: Discussionmentioning

confidence: 79%

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Fine structure in the large n limit of the non-Hermitian Penner matrix model

2015

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In this paper we apply results on the asymptotic zero distribution of the Laguerre polynomials to discuss generalizations of the standard large n limit in the non-hermitian Penner matrix model. In these generalizations g n n → t, but the product g n n is not necessarily fixed to the value of the 't Hooft coupling t. If t > 1 and the limit l = lim n→∞ | sin(π/g n )| 1/n exists, then the large n limit is well-defined but depends both on t and on l. This result implies that for t > 1 the standard large n limit with g n n = t fixed is not well-defined. The parameter l determines a fine structure of the asymptotic eigenvalue support: for l = 0 the support consists of an interval on the real axis with charge fraction Q = 1 − 1/t and an l-dependent oval around the origin with charge fraction 1/t. For l = 1 these two components meet, and for l = 0 the oval collapses to the origin. We also calculate the total electrostatic energy E, which turns out to be independent of l, and the free energy F = E − Q ln l, which does depend of the fine structure parameter l. The existence of large n asymptotic expansions of F beyond the planar limit as well as the double-scaling limit are also discussed.

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Section: Discussionmentioning

confidence: 79%

“…Riemann-Hilbert and steepest-descent methods [15,26,27,28] have permitted the complete characterization of the asymptotic zero distribution ρ L (z) of the scaled Laguerre polynomials L …”

Section: Zero Asymptotics Of Scaled Laguerre Polynomialsmentioning

confidence: 99%

Fine structure in the large n limit of the non-Hermitian Penner matrix model

2015

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“…, q} ⊂ N . Indeed, for t = 0, the left-hand member (7) reduces to ϕ and, thus, the support of the equilibrium measure starts at one or several of the critical points of the external field; in a similar fashion, these critical points of ϕ are the initial conditions of the dynamical system (11). Regarding the second question, we have, Theorem 2.2.…”

Section: F (X)mentioning

confidence: 92%

Equilibrium measures in the presence of certain rational external fields

Orive

Sánchez-Lara

2015

Journal of Mathematical Analysis and Applications

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Equilibrium measures in the real axis in the presence of rational external fields are considered. These external fields are called rational since their derivatives are rational functions. We analyze the evolution of the equilibrium measure, and its support, when the size of the measure, t, or other parameters in the external field vary. Our analysis is illustrated by studying with detail the case of a generalized Gauss-Penner model, which, in addition to its mathematical relevance, has important physical applications (in the framework of random matrix models). This paper is a natural continuation of [18], where equilibrium measures in the presence of polynomial external fields are thoroughly studied.

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“…For a point x 1 x 1 a complex chart can be assigned as follows: 14) where the branches of the radicals are taken such that ζ(w) > 0 when w is real such that w > 1 or w < 0. Similarly, to assign a complex chart to a point x 2 + ih 1 x 2 + ih, we use the following mapping: 15) with appropriate branches of the radicals. To a point of R corresponding to an infinite boundary point b 1 , a complex chart can be assigned via the function ζ = exp(−2πiw) for w such that 0 ≤ w ≤ 1, w < 0, (8.16) which maps the half-strip {w : 0 ≤ w ≤ 1, w < 0} onto the unit disc punctured at ζ = 0.…”

Section: Identifying Simple Critical Geodesics and Critical Loopsmentioning

confidence: 99%

“…Two main themes of this work are asymptotic behavior of zeros of certain polynomials and topological properties of related quadratic differentials. The study of asymptotic root distributions of hypergeometric, Jacobi, and Laguerre polynomials with variable real parameters, which grow linearly with degree, became a rather hot topic in recent publications, which attracted attention of many authors [14], [15], [16], [17], [18], [22], [24], [25], [27]. In this paper, we survey some known results in this area and present some new results keeping focus on Jacobi polynomials.…”

Section: Introduction: From Jacobi Polynomials To Quadratic Differentmentioning

confidence: 99%

Root-counting Measures of Jacobi Polynomials and Topological Types and Critical Geodesics of Related Quadratic Differentials

Shapiro

Solynin

2017

Trends in Mathematics

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Two main topics of this paper are asymptotic distributions of zeros of Jacobi polynomials and topology of critical trajectories of related quadratic differentials. First, we will discuss recent developments and some new results concerning the limit of the root-counting measures of these polynomials. In particular, we will show that the support of the limit measure sits on the critical trajectories of a quadratic differential of the form Q(z) dz 2 = az 2 +bz+c (z 2 −1) 2 dz 2 . Then we will give a complete classification, in terms of complex parameters a, b, and c, of possible topological types of critical geodesics for the quadratic differential of this type.

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The Szegő curve and Laguerre polynomials with large negative parameters

Abstract: We study the asymptotic zero distribution of the rescaled Laguerre polynomials, Lwith the parameter α n varying in such a way that lim n→∞ α n /n = −1. The connection with the so-called Szegő curve is shown.

Cited by 11 publications

References 22 publications

Fine structure in the large n limit of the non-Hermitian Penner matrix model

Fine structure in the large n limit of the non-Hermitian Penner matrix model

Equilibrium measures in the presence of certain rational external fields

Root-counting Measures of Jacobi Polynomials and Topological Types and Critical Geodesics of Related Quadratic Differentials

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