Determinantal Random Point Fields

The paper contains an exposition of recent as well as sufficiently old 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 determinantal random point field and a criterion for the weak convergence of its the distribution. In the second section we proceed with the examples of the determinantal random 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 last section we discuss the ∗permanent address