Determinantal random point fields

TL;DR

This paper reviews and presents key results on determinantal random point fields, including existence conditions, examples from various fields, and limit theorems for particle counts and spacings.

Contribution

It provides a comprehensive exposition of recent and classical results on determinantal point fields, including new characterizations and proofs of limit theorems.

Findings

Necessary and sufficient conditions for existence of determinantal fields

Characterization of fields with i.i.d. spacings in 1D and Z

Proof of CLT and functional CLT for particle counts

Abstract

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 R^1 and Z^1 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.