Trajectories of Quadratic Differentials for Jacobi Polynomials with Complex Parameters

Martínez–Finkelshtein, Andrei; Martinez-Gonzalez, P.; Thabet, Faouzi

doi:10.1007/s40315-015-0146-7

Cited by 16 publications

(12 citation statements)

References 18 publications

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“…These ideas have been successfully applied by Bertola [15], being closely related (in fact, in some sense equivalent) to the max-min approach mentioned before, see also [18]. In the same spirit and in a much more explicit form, quadratic differentials also appeared in similar asymptotic problems in [9,10,42,49,50,57]. We emphasize that in all these problems, the underlying quadratic differentials are defined on the complex plane.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 78%

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: 78%

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

Martínez–Finkelshtein

Silva

2016

Advances in Mathematics

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“…(see [36,41,42,44] for a general treatment of this case). In this situation the sequence P (α n ,β n ) n satisfies varying orthogonality conditions, when the weight in (2.7) depends on the degree of the polynomial.…”

Section: Cauchy Transforms and The Logarithmic Potential Theorymentioning

confidence: 99%

“…we can obtain the explicit expression for R using the arguments of Section 3. By the GRS theory, the problem of the weak-* asymptotics of the zeros of such polynomials boils down to the proof of the existence of a critical trajectory of the corresponding quadratic differential, joining the two zeros of C, and of the connectedness of its complement in C, see [36,41,42,44], as well as Figure 6.…”

Section: The Grs Theory and 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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“…Let γ ∈ J a(t),b(t) be the union of the part of γ a(t) from a(t) to P a(t) , the part of σ from P a(t) to P b(t) and the part of γ b(t) from P b(t) to b(t). Integrating along γ and using P a(t) a(t) P t (z) dz = b(t) P b(t) P t (z) dz = 0, [8] Finite critical trajectories 87 we find that…”

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confidence: 96%

On the Existence of Finite Critical Trajectories in a Family of Quadratic Differentials

Thabet

2016

Bull. Aust. Math. Soc.

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In this note, we discuss the possible existence of finite critical trajectories connecting two zeros in a family of quadratic differentials satisfying some properties. We treat the cases of holomorphic and meromorphic quadratic differentials. In addition, we reprove some results about the support of limiting root-counting measures of the generalized Laguerre and Jacobi polynomials with varying parameters.2010 Mathematics subject classification: 30C15, 31A35, 34E05.

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Trajectories of Quadratic Differentials for Jacobi Polynomials with Complex Parameters

Cited by 16 publications

References 18 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

Do Orthogonal Polynomials Dream of Symmetric Curves?

On the Existence of Finite Critical Trajectories in a Family of Quadratic Differentials

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