Discrete approximations of determinantal point processes on continuous spaces: tree representations and tail triviality

TL;DR

This paper proves tail triviality for determinantal point processes on continuous spaces by constructing discrete tree approximations, extending previous results from discrete to continuous settings, and highlighting implications for stochastic dynamics.

Contribution

The paper introduces a method to approximate continuous determinantal point processes with discrete tree representations, establishing tail triviality in the continuous case for the first time.

Findings

Tail triviality holds for determinantal point processes on continuous spaces.

Tree representations effectively approximate continuous processes.

Tail triviality ensures the equivalence of different stochastic dynamics constructions.

Abstract

We prove tail triviality of determinantal point processes $ \mu $ on continuous spaces. Tail triviality had been proved for such processes only on discrete spaces, and hence we have generalized the result to continuous spaces. To do this, we construct tree representations, that is, discrete approximations of determinantal point processes enjoying a determinantal structure. There are many interesting examples of determinantal point processes on continuous spaces such as zero points of the hyperbolic Gaussian analytic function with Bergman kernel, and the thermodynamic limit of eigenvalues of Gaussian random matrices for Sine$_2 $, Airy$_2 $, Bessel$_2 $, and Ginibre point processes. Tail triviality of $ \mu $ plays a significant role in proving the uniqueness of solutions of infinite-dimensional stochastic differential equations (ISDEs) associated with $ \mu $. For particle systems in $ \R $ arising from random matrix theory, there are two completely different constructions of natural stochastic dynamics. One is given by stochastic analysis through ISDEs and Dirichlet form theory, and the other is an algebraic method based on space-time correlation functions. Tail triviality is used crucially to prove the equivalence of these two stochastic dynamics.