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 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 .

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

“…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.…”

“…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).…”

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

“…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, . .…”

“…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.…”

On Meijer's $G$ function $G^{m,n}_{p,p}$ for $m+n=p$

“…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].…”