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

2016/12/06 by Tom Claeys, Claeys, Tom, Manuela Girotti +3
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1612.01916

openalex publication_date 2016/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the distribution of the smallest eigenvalue for certain classes of\npositive-definite Hermitian random matrices, in the limit where the size of the\nmatrices becomes large. Their limit distributions can be expressed as Fredholm\ndeterminants of integral operators associated to kernels built out of Meijer\nG-functions or Wright's generalized Bessel functions. They generalize in a\nnatural way the hard edge Bessel kernel Fredholm determinant. We express the\nlogarithmic derivatives of the Fredholm determinants identically in terms of a\n2\× 2 Riemann-Hilbert problem, and use this representation to obtain the\nso-called large gap asymptotics.\n

Citations

• Singular Values of Products of Ginibre Random Matrices

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