The asymptotic zero distribution of multiple orthogonal polynomials associated with Macdonald functions

Zhang, Lun; Román, Pablo

doi:10.1016/j.jat.2010.08.003

Cited by 10 publications

(9 citation statements)

References 28 publications

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“…The notion of a Nikishin system originated in works on Hermite-Padé rational approximation, see [5,56,62,76]. Vector equilibrium problems also played a role in the recent papers [8,15,17,65] that are related to random matrix theory and [6,43,44,69,84] that are related to recurrence relations. I(ν j , ν j+1 )…”

Section: Case α =mentioning

confidence: 99%

The Hermitian two matrix model with an even quartic potential

Duits

Kuijlaars

2012

Memoirs of the AMS

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We consider the two matrix model with an even quartic potential W (y) = y 4 /4 + αy 2 /2 and an even polynomial potential V (x). The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices M 1 . The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a 4 × 4 matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of M 1 . Our results generalize earlier results for the case α = 0, where the external field on the third measure was not present.

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Section: Case α =mentioning

confidence: 99%

The Hermitian two matrix model with an even quartic potential

Duits

Kuijlaars

2012

Memoirs of the AMS

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“…We expect that our main Theorems 2.13/2.15 and 3.1 may be used to obtain some results in the theory of multiple and matrix orthogonal polynomials on the real line. In fact, recently several papers appeared [8,13,23] which apply the results of Duits and Kuijlaars [9] on scalar banded Toeplitz matrices, to the context of multiple orthogonal polynomials. The recurrence relations of these polynomials lead to a banded Hessenberg matrix.…”

Section: Example 3: a Degenerate Case IImentioning

confidence: 99%

“…Typically this matrix is not exactly Toeplitz but only asymptotically. More generally, the orthogonality weights may be varying with n, which leads to so-called locally Toeplitz matrices [8,13,23]. We anticipate that more applications of this type may arise in the future, possibly leading to block (rather than scalar) Toeplitz matrices.…”

Section: Example 3: a Degenerate Case IImentioning

confidence: 99%

Equilibrium problem for the eigenvalues of banded block Toeplitz matrices

Delvaux

2012

Mathematische Nachrichten

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We consider banded block Toeplitz matrices Tn with n block rows and columns. We show that under certain technical assumptions, the normalized eigenvalue counting measure of Tn for n → ∞ weakly converges to one component of the unique vector of measures that minimizes a certain energy functional. In this way we generalize a recent result of Duits and Kuijlaars for the scalar case. Along the way we also obtain an equilibrium problem associated to an arbitrary algebraic curve, not necessarily related to a block Toeplitz matrix.For banded block Toeplitz matrices, there are several new phenomena that do not occur in the scalar case: (i) The total masses of the equilibrium measures do not necessarily form a simple arithmetic series but in general are obtained through a combinatorial rule; (ii) The limiting eigenvalue distribution may contain point masses, and there may be attracting point sources in the equilibrium problem; (iii) More seriously, there are examples where the connection between the limiting eigenvalue distribution of Tn and the solution to the equilibrium problem breaks down. We provide sufficient conditions guaranteeing that no such breakdown occurs; in particular we show this if Tn is a Hessenberg matrix.

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“…On the one hand, although the polynomials orthogonal with respect to the modified Bessel functions in the usual sense do not exhibit nice properties as the classical ones, the associated multiple orthogonal polynomials indeed have some interesting properties, and remarkably, they are all closely related to some stochastic models. The type I and type II multiple orthogonal polynomials associated with the modified Bessel functions of the second kind were introduced and studied in [7,11,14,42,45]. Their connections to the products of independent Ginibre matrices have been revealed recently in [33,44].…”

Section: Introductionmentioning

confidence: 99%

Mixed type multiple orthogonal polynomials associated with the modified Bessel functions and products of two coupled random matrices

Zhang

2017

Journal of Approximation Theory

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We consider mixed type multiple orthogonal polynomials associated with a system of weight functions consisting of two vectors. One vector is defined in terms of scaled modified Bessel function of the first kind I µ and I µ+1 , the other vector is defined in terms of scaled modified Bessel function of the second kind K ν and K ν+1 . We show that the corresponding mixed type multiple orthogonal polynomials exist. For the special case that each multi-index is on or close to the diagonal, basic properties of the polynomials and their linear forms are investigated, which include explicit formulas, integral representations, differential properties, limiting forms and recurrence relations. It comes out that, for specified parameters, the linear forms of these mixed type multiple orthogonal polynomials can be interpreted as biorthogonal functions encountering in recent studies of products of two coupled random matrices. This particularly implies a Riemann-Hilbert characterization of the correlation kernel, which provides an alternative way for further asymptotic analysis.

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The asymptotic zero distribution of multiple orthogonal polynomials associated with Macdonald functions

Cited by 10 publications

References 28 publications

The Hermitian two matrix model with an even quartic potential

The Hermitian two matrix model with an even quartic potential

Equilibrium problem for the eigenvalues of banded block Toeplitz matrices

Mixed type multiple orthogonal polynomials associated with the modified Bessel functions and products of two coupled random matrices

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