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date: 24 June 2019

Abstract and Keywords

This article focuses on the use of the orthogonal polynomial method for computing correlation functions, cluster functions, gap probability, Janossy density, and spacing distributions for the eigenvalues of matrix ensembles with unitary-invariant probability law. It first considers the classical families of orthogonal polynomials (Hermite, Laguerre, and Jacobi) and some corresponding unitary ensembles before discussing the statistical properties of N-tuples of real numbers. It then reviews the definitions of basic statistical quantities and demonstrates how their distributions can be made explicit in terms of orthogonal polynomials. It also describes the k-point correlation function, Fredholm determinants of finite-rank kernels, and resolvent kernels.

Keywords: orthogonal polynomial method, correlation function, cluster function, gap probability, Janossy density, spacing, orthogonal polynomial, unitary ensemble, Fredholm determinant, resolvent kernel

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