Extremal domains associated with an analytic function II

Stahl, Herbert

doi:10.1080/17476938508814118

Cited by 68 publications

(89 citation statements)

References 3 publications

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“…Orthogonal polynomials in which the weight can be a complex function and the integration path can be a general curve in the complex plane (nonhermitian orthogonal polynomials) first appeared in the mathematical literature as denominators of Padé and other types or rational approximants [23,24,25,26]. The theory quickly developed and found applications into such fields as the Riemann-Hilbert approach to strong asymptotics, random matrix theory [10,4,5,6,3,2] and in the study of dualities between supersymmetric gauge theories and string models [18,7,11,12,15].…”

Section: Introductionmentioning

confidence: 99%

“…This method is based on the existence of a unique normalized equilibrium charge density that minimizes the electrostatic energy (among all normalized charge densities supported on the curve Γ) in the presence of the external electrostatic potential V (z) = Re W (z) [21]. The method in [1] also relies heavily on the concepts of S-property and Scurve of Stahl [23,24,25,26] and of Gonchar and Rakhmanov [14,19,20], and on the fundamental result of Gonchar and Rakhmanov [14] asserting that if Γ is an S-curve, then the asymptotic zero density of P n (z) exists and is given by the equilibrium charge density of the associated electrostatic model. Note that the integral (1) is invariant under deformations of the curve Γ into curves in the same homology class and connecting the same two convergence sectors at infinity.…”

Section: Introductionmentioning

confidence: 99%

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Phase structure and asymptotic zero densities of orthogonal polynomials in the cubic model

Álvarez

Alonso

Медина

2015

Journal of Computational and Applied Mathematics

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Phase structure and asymptotic zero densities of orthogonal polynomials in the cubic model

Álvarez

Alonso

Медина

2015

Journal of Computational and Applied Mathematics

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“…This equilibrium property is the so-called S-property of Stahl [25][26][27][28] and of Gonchar and Rakhmanov [29,30].…”

Section: The Electrostatic Interpretationmentioning

confidence: 99%

Complex saddle points in the Gross-Witten-Wadia matrix model

2016

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We give an exhaustive characterization of the complex saddle point configurations of the Gross-WittenWadia matrix model in the large-N limit. In particular, we characterize the cases in which the saddles accumulate in one, two, or three arcs, in terms of the values of the coupling constant and of the fraction of the total unit density that is supported in one of the arcs, and derive an explicit condition for gap closing associated with nonvacuum saddles. By applying the idea of large-N instanton we also give direct analytic derivations of the weak-coupling and strong-coupling instanton actions.

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“…Equation (37) is the so-called S-property of Stahl [21,22,23] and of Gonchar and Rakhmanov [24,25], whose electrostatic interpretation is that the electric fields at either side of γ are opposite or, equivalently, that the forces acting on each element of charge at z from the two sides of γ are equal. The total electrostatic energy (8) can be written in terms of the external potential and the constants u i as…”

Section: Critical Densitiesmentioning

confidence: 99%

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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Extremal domains associated with an analytic function II

Cited by 68 publications

References 3 publications

Phase structure and asymptotic zero densities of orthogonal polynomials in the cubic model

Phase structure and asymptotic zero densities of orthogonal polynomials in the cubic model

Complex saddle points in the Gross-Witten-Wadia matrix model

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

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