Local universality in biorthogonal Laguerre ensembles

Abstract

We consider $n$ particles $0\le x_1<x_2<\cdots <x_n<+\infty$, distributed according to a probability measure of the form $$ \frac{1}{Z_n}\prod_{1\leq i <j \leq n}(x_j-x_i)\prod_{1\leq i <j \leq n}(x_j^θ-x_i^θ)\prod_{j=1}^nx_j^αe^{-x_j}\ud x_j, ~~ α>-1,~~ θ>0, $$ where $Z_n$ is the normalization constant. This distribution arises in the context of modeling disordered conductors in the metallic regime, and can also be realized as the distribution for squared singular values of certain triangular random matrices. We give a double contour integral formula for the correlation kernel, which allows us to establish universality for the local statistics of the particles, namely, the bulk universality and the soft edge universality via the sine kernel and the Airy kernel, respectively. In particular, our analysis also leads to new double contour integral representations of scaling limits at the origin (hard edge), which are equivalent to those found in the classical work of Borodin. We conclude this paper by relating the correlation kernels to those appearing in recent studies of products of $M$ Ginibre matrices for the special cases $\theta =M\in \mathbb{N}$.

25 pages, 3 figures, revised version according to the suggestions of the anonymous referees. To appear in Journal of Statistical Physics

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Cited by in corpus (19)

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• CLT for biorthogonal ensembles and related combinatorial identities
• Singular values for products of complex Ginibre matrices with a source: hard edge limit and phase transition
• Muttalib--Borodin ensembles in random matrix theory --- realisations and correlation functions
• The local universality of Muttalib-Borodin biorthogonal ensembles with parameter $\theta =\frac{1}{2}$
• On Wright's generalized Bessel kernel
• Matrix product ensembles of Hermite-type and the hyperbolic Harish-Chandra-Itzykson-Zuber integral
• Bulk and soft-edge universality for singular values of products of Ginibre random matrices
• $d$-Orthogonal Analogs of Classical Orthogonal Polynomials
• Generalized random matrix model with additional interactions
• The local universality of Muttalib-Borodin ensembles when the parameter $\theta$ is the reciprocal of an integer
• Gaussian perturbations of hard edge random matrix ensembles
• Relations between generalised Wishart matrices, the Muttalib--Borodin model and matrix spherical functions
• Nonmonotonic confining potential and eigenvalue density transition for generalized random matrix model
• Global rigidity and exponential moments for soft and hard edge point processes
• Rate of convergence at the hard edge for various Pólya ensembles of positive definite matrices
• Large gap asymptotics at the hard edge for product random matrices and Muttalib-Borodin ensembles
• On the computation of density and two-point correlation functions of a class of random matrix ensembles
• Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles