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

Abstract

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.

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

“…It comes out that the conditions (1.4) and (1.21) ensure the potential function V is one-cut regular, as shown in in [39,Proposition 3.6] for θ = 1/2, and later confirmed in [46] for all rational θ. Here we note that the argument in [39] works for all θ > 0; see also [21,Theorem 1.8] for a slightly weaker sufficient condition for the one-cut regular property.…”

“…Moreover, the equilibrium measure µ = µ (V ) is characterized by the following Euler-Lagrange conditions: log|x − y|dµ(y) + log|x θ − y θ |dµ(y) − V (x) = , x ∈ supp(µ), (1.6) log|x − y|dµ(y) + log|x θ − y θ |dµ(y) − V (x) ≤ , x ∈ [0, +∞), (1.7) where is some real constant. Following [39,46], we require the potential V to be one-cut regular, in the sense that…”

“…The scaling limit of correlation kernel then follows from asymptotic analysis of the associated (r + 1) × (r + 1) RH problem [55]. This is exactly the strategy used in [39,46]. If θ ∈ N, this idea, however, is not applicable, since it is not clear what the large size RH problem would look like; see also the comments after [46,Theorem 1.1] about the limitations of using the transformation x → x 1/θ to achieve the goal.…”

“…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.…”