Partition functions and the continuum limit in Penner matrix models

Álvarez, Gabriel; Alonso, Luis Martı́nez; Медина, Елена

doi:10.1088/1751-8113/47/31/315205

Cited by 16 publications

(62 citation statements)

References 47 publications

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“…Recently, non-hermitian orthogonality found its way to areas with a much more "modern" flavor, playing the crucial role in the description of the rational solutions to Painlevé equations [12,17], in theoretical physics [1,2,20,25] and in numerical analysis [27].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Critical measures for vector energy: Global structure of trajectories of quadratic differentials

Martínez–Finkelshtein

Silva

2016

Advances in Mathematics

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Abstract. Saddle points of a vector logarithmic energy with a vector polynomial external field on the plane constitute the vector-valued critical measures, a notion that finds a natural motivation in several branches of analysis. We study in depth the case of measures µ = (µ 1 , µ 2 , µ 3 ) when the mutual interaction comprises both attracting and repelling forces.For arbitrary vector polynomial external fields we establish general structural results about critical measures, such as their characterization in terms of an algebraic equation solved by an appropriate combination of their Cauchy transforms, and the symmetry properties (or the S-properties) exhibited by such measures. In consequence, we conclude that vector-valued critical measures are supported on a finite number of analytic arcs, that are trajectories of a quadratic differential globally defined on a three-sheeted Riemann surface. The complete description of the so-called critical graph for such a differential is the key to the construction of the critical measures.We illustrate these connections studying in depth a one-parameter family of critical measures under the action of a cubic external field. This choice is motivated by the asymptotic analysis of a family of (non-hermitian) multiple orthogonal polynomials, that is subject of a forthcoming paper. Here we compute explicitly the Riemann surface and the corresponding quadratic differential, and analyze the dynamics of its critical graph as a function of the parameter, giving a detailed description of the occurring phase transitions. When projected back to the complex plane, this construction gives us the complete family of vector-valued critical measures, that in this context turn out to be vector-valued equilibrium measures.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Critical measures for vector energy: Global structure of trajectories of quadratic differentials

Martínez–Finkelshtein

Silva

2016

Advances in Mathematics

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“…This partition function depends implicitly on λ, although for simplicity of notation we do not emphasize it. Ensembles of the form (1.7), with extra algebraic terms in the density, were studied extensively in the literature on matrix models under the name Gauss-Penner model, see [20,2] for details and applications.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the probability of positive-definiteness in the gGUE via semi-classical Laguerre polynomials

Deaño

Simm

2017

Journal of Approximation Theory

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A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription.For more information, please contact the WRAP Team at: wrap@warwick.ac.ukOn the probability of positive-definiteness in the gGUE via semi-classical Laguerre polynomials Alfredo Deaño * and Nicholas J. Simm † May 5, 2017AbstractIn this paper, we compute the probability that an N × N matrix from the generalised Gaussian Unitary Ensemble (gGUE) is positive definite, extending a previous result of Dean and Majumdar [15]. For this purpose, we work out the large degree asymptotics of semi-classical Laguerre polynomials and their recurrence coefficients, using the steepest descent analysis of the corresponding Riemann-Hilbert problem.

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“…(6.8) 2 As a referee pointed out, the configuration of Figure 7, top right, could be formally considered as a limit case of what we describe in Theorem 6.1 below, where the S-property does play a role. Nevertheless, the situation is completely different here due to the existence of a string of zeros of the weight (6.6) along the imaginary axis.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…where R is a polynomial and D a rational function with poles of order at most 2. This fact yields that measures µ 1 , µ 2 , µ 3 are supported on a finite union of analytic arcs, that are trajectories of a quadratic differential living on the Riemann surface of (5.6), and which is explicitly given on each sheet of this Riemann surface as (ξ i − ξ j ) 2 (z)(dz) 2 . The fact that the support of critical (or equilibrium) vector measures is described in terms of trajectories on a compact Riemann surface (and thus, what we actually see is their projection on the plane) explains the apparent geometric complexity of the limit set for the zeros of Hermite-Padé polynomials, see Figure 4.…”

Section: Vector Critical Measuresmentioning

confidence: 99%

Do Orthogonal Polynomials Dream of Symmetric Curves?

Martínez–Finkelshtein

Rakhmanov

2016

Found Comput Math

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Abstract. The complex or non-Hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the freedom in the choice of the integration contour for such polynomials, the location of their zeros is a priori not clear. Nevertheless, numerical experiments, such as those presented in this paper, show that the zeros not simply cluster somewhere on the plane, but persistently choose to align on certain curves, and in a very regular fashion.The problem of the limit zero distribution for the non-Hermitian orthogonal polynomials is one of the central aspects of their theory. Several important results in this direction have been obtained, especially in the last 30 years, and describing them is one of the goals of the first parts of this paper. However, the general theory is far from being complete, and many natural questions remain unanswered or have only a partial explanation.Thus, the second motivation of this paper is to discuss some "mysterious" configurations of zeros of polynomials, defined by an orthogonality condition with respect to a sum of exponential functions on the plane, that appeared as a results of our numerical experiments. In this apparently simple situation the zeros of these orthogonal polynomials may exhibit different behaviors: for some of them we state the rigorous results, while others are presented as conjectures (apparently, within a reach of modern techniques). Finally, there are cases for which it is not yet clear how to explain our numerical results, and where we cannot go beyond an empirical discussion.

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Partition functions and the continuum limit in Penner matrix models

Cited by 16 publications

References 47 publications

Critical measures for vector energy: Global structure of trajectories of quadratic differentials

Critical measures for vector energy: Global structure of trajectories of quadratic differentials

On the probability of positive-definiteness in the gGUE via semi-classical Laguerre polynomials

Do Orthogonal Polynomials Dream of Symmetric Curves?

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