2019
DOI: 10.1088/1361-6544/ab247c
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The local universality of Muttalib–Borodin biorthogonal ensembles with parameter $\theta = \frac{1}{2}$

Abstract: 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 ing… Show more

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Cited by 28 publications
(83 citation statements)
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“…Moreover, from the behavior of μ described above, one expects the sine kernel in the bulk and the Airy kernel at the soft edge for a large class of weights. This has been proved in the special case θ = 1 2 only recently by Kuijlaars and Molag in [24] using a non-standard analysis of a 3 × 3 matrix RH problem. The case of general θ is still open.…”
Section: Introduction and Main Resultsmentioning
confidence: 89%
See 1 more Smart Citation
“…Moreover, from the behavior of μ described above, one expects the sine kernel in the bulk and the Airy kernel at the soft edge for a large class of weights. This has been proved in the special case θ = 1 2 only recently by Kuijlaars and Molag in [24] using a non-standard analysis of a 3 × 3 matrix RH problem. The case of general θ is still open.…”
Section: Introduction and Main Resultsmentioning
confidence: 89%
“…with χ γ and χγ denoting the indicator functions of γ andγ , respectively. Using some results from [3,4] and following the procedure developed by Its-Izergin-Korepin-Slavnov (IIKS) [19], the authors of [12] obtained a differential identity for 24) in terms of the solution Y of a 2 × 2 matrix RH problem. Moreover, by performing a (non-standard) Deift/Zhou [17] steepest descent analysis of this RH problem, they computed the large s asymptotics of the expression in (1.24).…”
Section: Outline Of Proofsmentioning
confidence: 99%
“…Inspired by the principle of universality, a fundamental issue in random matrix theory, it is generally conjectured that, as n → ∞, the local behaviours of the associated biorthogonal polynomials and correlation kernel are the same as those derived for classical weights for a large class of V with the parameter θ > 0 fixed. An attempt to justify this conjecture is given in recent works [39,46], where local universality of the correlation kernel near the origin is established for 1/θ ∈ N = {1, 2, . .…”
Section: The Modelmentioning
confidence: 99%
“…under weak conditions on V . It is the aim of the present work to prove a parallel universality result for a general class of V and θ ∈ N. Moreover, our approach is different from that used in [39,46], which might be of independent interest and serves as the other main contribution of this paper. We next state our main results.…”
Section: The Modelmentioning
confidence: 99%
“…In this paper, however, we will restrict our attention to the principal value of the argument −π < arg(z) ≤ π. G function is a natural generalization of the generalized hypergeometric function and is indispensable when constructing the fundamental solutions of the generalized hypergeometric differential equation [8,17,19]. It is also important in probability and statistics [6,21], random matrix theory and in adjacent field of multiple orthogonal polynomials [2,13].…”
Section: Introductionmentioning
confidence: 99%