Large gap asymptotics at the hard edge for product random matrices and Muttalib-Borodin ensembles

Abstract

We study the distribution of the smallest eigenvalue for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. Their limit distributions can be expressed as Fredholm determinants of integral operators associated to kernels built out of Meijer $G$-functions or Wright's generalized Bessel functions. They generalize in a natural way the hard edge Bessel kernel Fredholm determinant. We express the logarithmic derivatives of the Fredholm determinants identically in terms of a $2\times 2$ Riemann-Hilbert problem, and use this representation to obtain the so-called large gap asymptotics.

arXiv comment41 pages, 6 figures

References in corpus (11)

• Level-Spacing Distributions and the Airy Kernel
• Biorthogonal ensembles
• Products of Rectangular Random Matrices: Singular Values and Progressive Scattering
• Singular value statistics of matrix products with truncated unitary matrices
• The Cauchy two-matrix model
• The transition between the gap probabilities from the Pearcey to the Airy process; a Riemann-Hilbert approach
• Differential equations for singular values of products of Ginibre random matrices
• Local universality in biorthogonal Laguerre ensembles
• Muttalib--Borodin ensembles in random matrix theory --- realisations and correlation functions
• On Wright's generalized Bessel kernel
• Singular Values of Products of Ginibre Random Matrices

Cited by in corpus (1)

• Integrable structure of products of finite complex Ginibre random matrices