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Uspekhi Mat. Nauk, 2000, Volume 55, Issue 5(335), Pages 107–160 (Mi umn321)  

This article is cited in 287 scientific papers (total in 287 papers)

Determinantal random point fields

A. B. Soshnikov

University of California, Berkeley

Abstract: This paper contains an exposition of both recent and rather old results on determinantal random point fields. We begin with some general theorems including proofs of necessary and sufficient conditions for the existence of a determinantal random point field with Hermitian kernel and of a criterion for weak convergence of its distribution. In the second section we proceed with examples of determinantal random fields in quantum mechanics, statistical mechanics, random matrix theory, probability theory, representation theory, and ergodic theory. In connection with the theory of renewal processes, we characterize all Hermitian determinantal random point fields on $\mathbb R^1$ and $\mathbb Z^1$ with independent identically distributed spacings. In the third section we study translation-invariant determinantal random point fields and prove the mixing property for arbitrary multiplicity and the absolute continuity of the spectra. In the last section we discuss proofs of the central limit theorem for the number of particles in a growing box and of the functional central limit theorem for the empirical distribution function of spacings.


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English version:
Russian Mathematical Surveys, 2000, 55:5, 923–975

Bibliographic databases:

UDC: 519.218
MSC: Primary 60G60, 60K35; Secondary 15A52, 60K05, 60F05, 37A25, 82C22
Received: 11.04.2000

Citation: A. B. Soshnikov, “Determinantal random point fields”, Uspekhi Mat. Nauk, 55:5(335) (2000), 107–160; Russian Math. Surveys, 55:5 (2000), 923–975

Citation in format AMSBIB
\by A.~B.~Soshnikov
\paper Determinantal random point fields
\jour Uspekhi Mat. Nauk
\yr 2000
\vol 55
\issue 5(335)
\pages 107--160
\jour Russian Math. Surveys
\yr 2000
\vol 55
\issue 5
\pages 923--975

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