A vector Riemann-Hilbert approach to the Muttalib-Borodin ensembles

Wang, Dong; Zhang, Lun

doi:10.1016/j.jfa.2021.109380

Cited by 5 publications

(21 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since then, various extensions have been made, including to the asymptotics of orthogonal polynomials. More specifically, the extension on the unit circle was studied in [3], general measures and universality were studied in [10,17,24,30,53,56], the biorthogonal polynomial problem was studied in [8,49,55,75] and multiple orthogonal polynomials were studied in [74]. For a more comprehensive review, we refer the reader to [4,12,18,19,54].…”

Section: 1mentioning

confidence: 99%

See 1 more Smart Citation

A Riemann--Hilbert approach to the perturbation theory for orthogonal polynomials: Applications to numerical linear algebra and random matrix theory

Ding¹,

Trogdon²

2021

Preprint

View full text Add to dashboard Cite

We establish a new perturbation theory for orthogonal polynomials using a Riemann-Hilbert approach and consider applications in numerical linear algebra and random matrix theory. We show that the orthogonal polynomials with respect to two measures can be effectively compared using the difference of their Stieltjes transforms on a suitably chosen contour. Moreover, when two measures are close and satisfy some regularity conditions, we use the theta functions of a hyperelliptic Riemann surface to derive explicit and accurate expansion formulae for the perturbed orthogonal polynomials. The leading error terms can be fully characterized by the difference of the Stieltjes transforms on the contour. The results are applied to analyze several numerical algorithms from linear algebra, including the Lanczos tridiagonalization procedure, the Cholesky factorization and the conjugate gradient algorithm (CGA). As a case study, we investigate these algorithms applied to a general spiked sample covariance matrix model by considering the eigenvector empirical spectral distribution and its limit, allowing for precise estimates on the algorithms as the number of iterations diverges. For this concrete random matrix model, beyond the first order expansion, we derive a mesoscopic central limit theorem for the associated orthogonal polynomials and other quantities relevant to numerical algorithms.

show abstract

Section: 1mentioning

confidence: 99%

“…We refer the readers to [12,18,54] for a more exhaustive discussion. The methodology has also applied to various random matrix models, see [9,11,15,20,21,36,57,58,75], to name but a few.…”

Section: 1mentioning

confidence: 99%

A Riemann--Hilbert approach to the perturbation theory for orthogonal polynomials: Applications to numerical linear algebra and random matrix theory

Ding¹,

Trogdon²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Thus, it is understood in the present day that the matrix product ensembles reviewed in this section, together with their approximations in terms of Muttalib-Borodin ensembles, belong to a universality class of eigenvalue ensembles that are characterised by having their bulk, soft edge, and (doublesided) hard edge statistics described respectively by the sine kernel, the Airy kernel, and the Meijer G-kernel with various parametrisations. This universality class has consequently garnered great interest, with recent works showing that it contains other matrix product ensembles [122], [216], [205], along with the class of Muttalib-Borodin ensembles with θ > 0 real and general weight of the form w(λ) = λ a e −NV(λ) χ λ>0 [215], [247], [311] (see also [67], [60] for related results on this latter class of Muttalib-Borodin ensembles).…”

Section: Relations To Matrix Product Ensemblesmentioning

confidence: 99%

Recursive Characterisations of Random Matrix Ensembles and Associated Combinatorial Objects

Rahman

2023

Bull. Aust. Math. Soc.

View full text Add to dashboard Cite

“…The advantage of this approach is that the framework does not depend on the value of θ, and it in principle works for all real positive θ. Practically, all the arguments in [55], except for the construction of the local parametrices at 0, are applicable for all real positive θ. In [55] the local parametrices at 0 were constructed for θ = m, m ∈ Z + , and they were in the form of (m + 1) × (m + 1) matrices.…”

Section: Novelty Of the Papermentioning

confidence: 99%

“…Practically, all the arguments in [55], except for the construction of the local parametrices at 0, are applicable for all real positive θ. In [55] the local parametrices at 0 were constructed for θ = m, m ∈ Z + , and they were in the form of (m + 1) × (m + 1) matrices. Thus, [55] localizes the Riemann-Hilbert analysis of the biorthogonal polynomials to a local parametrix construction problem, and it calls for new ideas to construct the local parametrices at 0 for general θ.…”

Section: Novelty Of the Papermentioning

confidence: 99%

The local universality of Muttalib–Borodin ensembles when the parameter θ is the reciprocal of an integer

Leslie

2021

Nonlinearity

View full text Add to dashboard Cite

The Muttalib–Borodin ensemble is a probability density function for n particles on the positive real axis that depends on a parameter θ and a weight w. We consider a varying exponential weight that depends on an external field V. In a recent article, the large n behavior of the associated correlation kernel at the hard edge was found for θ = 1 2 , where only few restrictions are imposed on V. In the current article we generalize the techniques and results of this article to obtain analogous results for θ = 1 r , where r is a positive integer. The approach is to relate the ensemble to a type II multiple orthogonal polynomial ensemble with r weights, which can then be related to an (r + 1) × (r + 1) Riemann–Hilbert problem. The local parametrix around the origin is constructed using Meijer G-functions. We match the local parametrix around the origin with the global parametrix with a double matching, a technique that was recently introduced.

show abstract

A vector Riemann-Hilbert approach to the Muttalib-Borodin ensembles

Cited by 5 publications

References 49 publications

A Riemann--Hilbert approach to the perturbation theory for orthogonal polynomials: Applications to numerical linear algebra and random matrix theory

A Riemann--Hilbert approach to the perturbation theory for orthogonal polynomials: Applications to numerical linear algebra and random matrix theory

Recursive Characterisations of Random Matrix Ensembles and Associated Combinatorial Objects

The local universality of Muttalib–Borodin ensembles when the parameter θ is the reciprocal of an integer

Contact Info

Product

Resources

About