2000/02/29 by Alexander Soshnikov, A Soshnikov · 8 citations
Mathematics · Physics and Astronomy · #Geometry and complex manifolds #Random Matrices and Applications #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR
paper · pdf · doi:10.1070/rm2000v055n05abeh000321
To appear in the Russian Mathematical Surveys; small misprints are corrected
crossref issued 2000/10/31 · crossref published 2000/10/31 · crossref published-print 2000/10/31 · openalex publication_date 2000/10/31 · arxiv created 2000/11/28 · crossref created 2002/08/24 · crossref published-online 2007/10/17 · arxiv updated 2015/06/26 · openalex created_date 2016/06/24 · crossref deposited 2025/05/12 · crossref indexed 2026/07/31 · openalex updated_date 2026/08/01
The paper contains an exposition of recent as well as old enough results on determinantal random point fields. We start with some general theorems including the proofs of the necessary and sufficient condition for the existence of the determinantal random point field with Hermitian kernel and a criterion for the weak convergence of its distribution. In the second section we proceed with the examples of the determinantal random point fields from 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 determinantal random point fields in R1 and Z1 with independent identically distributed spacings. In the third section we study the translation invariant determinantal random point fields and prove the mixing property of any multiplicity and the absolute continuity of the spectra. In the fourth (and the last) section we discuss the proofs of the Central Limit Theorem for the number of particles in the growing box and the Functional Central Limit Theorem for the empirical distribution function of spacings.