Painleve formulas of the limiting distributions for non-null complex sample covariance matrices

2005/04/29 by Jinho Baik, Baik, Jinho
Mathematics · #33E17 #60E99 #62E99 #Advanced Combinatorial Mathematics #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.math/0504606

openalex publication_date 2005/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In a recent study of large non-null sample covariance matrices, a new sequence of functions generalizing the GUE Tracy-Widom distribution of random matrix theory was obtained. This paper derives Painlevé formulas of these functions and use them to prove that they are indeed distribution functions. Applications of these new distribution functions to last passage percolation, queues in tandem and totally asymmetric simple exclusion process are also discussed. As a part of the proof, a representation of orthogonal polynomials on the unit circle in terms of an operator on a discrete set is presented.

Citations

• On the distribution of the largest eigenvalue in principal components analysis
• Eigenvalues of Large Sample Covariance Matrices of Spiked Population Models

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• Universality of the distribution functions of random matrix theory
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