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Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices. (English) Zbl 1139.33006
Author’s abstract: In a recent study of large nonnull sample covariance matrices, a new sequence of functions generalizing the Gaussian unitary ensemble (GUE) Tracy-Widom distribution of random matrix theory was obtained. This article derives Painlevé formulas of these functions and uses 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.

##### MSC:
 33E17 Painlevé-type functions 60E99 Distribution theory 62E99 Statistical distribution theory 15B52 Random matrices (algebraic aspects) 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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