A Vector Equilibrium Problem for Muttalib-Borodin Biorthogonal Ensembles

Kuijlaars, Arno B. J.

doi:10.3842/sigma.2016.065

Cited by 11 publications

(28 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The first term of the expansion (4.20) for g 2 comes from the fact that ν has total mass 1/2. The full expansion can be obtained from the proofs of Propositions 3.1 and 3.2 in [26]. Namely, in the proof of Proposition 3.2 we find the relation…”

Section: Second Transformation X → Tmentioning

confidence: 92%

“…The solution to the vector equilibrium problem [26], provides us with two measures µ * and ν * satisfying the variational conditions (2.8) and (2.9). Here µ * = µ * V, 1 2 is the θ = 1 2 -equilibrium measure in the external field V , which has a compact support supp(µ * ) = ∆ 1 , ∆ 1 = (0, q) (4.12) by our assumption on the external field V in Theorem 1.2.…”

Section: Second Transformation X → Tmentioning

confidence: 99%

See 1 more Smart Citation

The local universality of Muttalib–Borodin biorthogonal ensembles with parameter $\theta = \frac{1}{2}$

Kuijlaars

Leslie

2019

Nonlinearity

View full text Add to dashboard Cite

The Muttalib-Borodin biorthogonal ensemble is a probability density function for n particles on the positive real line that depends on a parameter θ and an external field V . For θ = 1 2 we find the large n behavior of the associated correlation kernel with only few restrictions on V . The idea is to relate the ensemble to a type II multiple orthogonal polynomial ensemble that can in turn be related to a 3 × 3 Riemann-Hilbert problem which we then solve with the Deift-Zhou steepest descent method. The main ingredient is the construction of the local parametrix at the origin, with the help of Meijer G-functions, and its matching condition with a global parametrix. We will present a new iterative technique to obtain the matching condition, which we expect to be applicable in more general situations as well.with limiting kernel.is Wright's generalization of the Bessel function. For θ = 1, the limit (1.5) reduces to the well-known Bessel kernel in the theory of random matrices. Several other expressions are known for the kernel (1.6). For example, Zhang [40, Theorem 1.2] gives a double contour integral, and if θ or 1/θ is an integer, then (1.5) can be expressed in terms of Meijer G-functions [30]. These so-called Meijer G-kernels appear in singular value distributions for products of random matrices [2, 3] and in their hard edge scaling limit [4,5,23,24,25,31]. See [1] for a survey paper. The bulk and soft edge scaling limits for singular values of Ginibre random matrices are the usual sine and Airy kernels [32], and these classical limits were also established for the Muttalib-Borodin model in the Laguerre case [40].It is natural to expect that the limit (1.5) is not restricted to the case V (x) = x, but holds for much more general external fields. In this paper we consider θ = 1 2 and we show that the hard edge scaling limit (1.5) indeed holds for a large class of external fields V .

show abstract

Section: Second Transformation X → Tmentioning

confidence: 92%

Section: Second Transformation X → Tmentioning

confidence: 99%

The local universality of Muttalib–Borodin biorthogonal ensembles with parameter $\theta = \frac{1}{2}$

Kuijlaars

Leslie

2019

Nonlinearity

View full text Add to dashboard Cite

show abstract

“…On the other hand, near a soft edge, this density vanishes to the order 1/2 for any value of θ ; this is the usual square root behavior that is often encountered in random matrix theory. We also refer to [5,8,23] for related results on the equilibrium measure.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Higher Order Large Gap Asymptotics at the Hard Edge for Muttalib–Borodin Ensembles

Charlier

Lenells

Mauersberger

2021

Commun. Math. Phys.

View full text Add to dashboard Cite

We consider the limiting process that arises at the hard edge of Muttalib–Borodin ensembles. This point process depends on $$\theta > 0$$ θ > 0 and has a kernel built out of Wright’s generalized Bessel functions. In a recent paper, Claeys, Girotti and Stivigny have established first and second order asymptotics for large gap probabilities in these ensembles. These asymptotics take the form $$\begin{aligned} {\mathbb {P}}(\text{ gap } \text{ on } [0,s]) = C \exp \left( -a s^{2\rho } + b s^{\rho } + c \ln s \right) (1 + o(1)) \qquad \text{ as } s \rightarrow + \infty , \end{aligned}$$ P ( gap on [ 0 , s ] ) = C exp - a s 2 ρ + b s ρ + c ln s ( 1 + o ( 1 ) ) as s → + ∞ , where the constants $$\rho $$ ρ , a, and b have been derived explicitly via a differential identity in s and the analysis of a Riemann–Hilbert problem. Their method can be used to evaluate c (with more efforts), but does not allow for the evaluation of C. In this work, we obtain expressions for the constants c and C by employing a differential identity in $$\theta $$ θ . When $$\theta $$ θ is rational, we find that C can be expressed in terms of Barnes’ G-function. We also show that the asymptotic formula can be extended to all orders in s.

show abstract

“…We do not believe it to be reasonable to expect that our method can be generalized to all θ > 0, but it might be possible to adapt it to obtain the same results for rational θ. In particular, it was shown in [30] that there exists an underlying vector equilibrium problem when θ is rational. The measures of its solution might be used to construct g-functions for a corresponding RHP, although, at the moment, it is not clear to us what this RHP would look like.…”

Section: Statement Of Resultsmentioning

confidence: 99%

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

Molag

2020

Preprint

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 Gfunctions. 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 Equilibrium Problem for Muttalib-Borodin Biorthogonal Ensembles

Cited by 11 publications

References 23 publications

The local universality of Muttalib–Borodin biorthogonal ensembles with parameter $\theta = \frac{1}{2}$

The local universality of Muttalib–Borodin biorthogonal ensembles with parameter $\theta = \frac{1}{2}$

Higher Order Large Gap Asymptotics at the Hard Edge for Muttalib–Borodin Ensembles

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

Contact Info

Product

Resources

About