On asymptotic zero distribution of Laguerre and generalized Bessel polynomials with varying parameters

Abstract. We study the asymptotic behavior of Laguerre polynomials L (αn) n (nz) as n → ∞, where α n is a sequence of negative parameters such that −α n /n tends to a limit A > 1 as n → ∞. These polynomials satisfy a non-hermitian orthogonality on certain contours in the complex plane. This fact allows the formulation of a Riemann-Hilbert problem whose solution is given in terms of these Laguerre polynomials. The asymptotic analysis of the Riemann-Hilbert problem is carried out by the steepest descent method of Deift and Zhou, in the same spirit as done by Deift et al. for the case of orthogonal polynomials on the real line. A main feature of the present paper is the choice of the correct contour.

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“…The correct contour was described in [20]. It is a curve with the S-property of Stahl [27] and Gonchar and Rakhmanov [16].…”

Section: Asymptotics From Riemann-hilbert Problemsmentioning

confidence: 99%

Riemann-Hilbert Analysis for Laguerre Polynomials with Large Negative Parameter

Kuijlaars

McLaughlin

2001

Comput. Methods Funct. Theory

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“…Note also how in both cases 25 out of the 60 zeros are on the real interval, i.e., a fraction 25/60 ≈ 0.417 already quite close to the limiting value 1 − 1/t = 1 − 1/ √ 3 ≈ 0.423. Finally, we notice that the results of [26] on the asymptotic zero distribution of Laguerre polynomials for values −∞ < A < −1 of the limit (48) can also be applied to the non-hermitian Penner model. These results imply the existence of an asymptotic eigenvalue density with a well-defined support for any sequence g n of coupling constants with limit (4) such that 0 < t < 1.…”

Section: Let Us Suppose That the Fixed 'T Hooft Coupling T Takes A Ramentioning

confidence: 88%

“…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%

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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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“…This means that the Laguerre polynomials are no longer orthogonal with respect to the weight function on the positive real axis, when α < −1. Recently, there is a growing interest in studying the behavior of the zeros of these polynomials, when α is a negative parameter and tends to −∞ as n → ∞; see [7,13] and also [8,12]. Here, we restrict our attention to the case α = α n = −A n n, (1.6) where {A n } is a sequence of constants tending to a limit A > 1 as n → ∞.…”

Section: Introductionmentioning

confidence: 99%

Global asymptotics for Laguerre polynomials with large negative parameter—a Riemann-Hilbert approach

Dai

Wong

2008

Ramanujan J

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In this paper, we study the asymptotic behavior of the Laguerre polynomials L (α n ) n (nz) as n → ∞. Here α n is a sequence of negative numbers and −α n /n tends to a limit A > 1 as n → ∞. An asymptotic expansion is obtained, which is uniformly valid in the upper half plane C + = {z : Im z ≥ 0}. A corresponding expansion is also given for the lower half plane C − = {z : Im z ≤ 0}. The two expansions hold, in particular, in regions containing the curve in the complex plane, on which these polynomials are orthogonal. Our method is based on the Riemann-Hilbert approach introduced by Deift and Zhou.

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On asymptotic zero distribution of Laguerre and generalized Bessel polynomials with varying parameters

Cited by 39 publications

References 9 publications

Riemann-Hilbert Analysis for Laguerre Polynomials with Large Negative Parameter

Riemann-Hilbert Analysis for Laguerre Polynomials with Large Negative Parameter

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

Global asymptotics for Laguerre polynomials with large negative parameter—a Riemann-Hilbert approach

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