Infinite Determinantal Measures

TL;DR

This paper introduces infinite determinantal measures as limits of determinantal measures and describes their role in the ergodic decomposition of infinite Pickrell measures on complex matrices.

Contribution

It provides an explicit description of ergodic decomposition for infinite Pickrell measures using infinite determinantal measures derived from finite-rank perturbations of Bessel processes.

Findings

Explicit description of ergodic decomposition for infinite Pickrell measures.

Representation of infinite determinantal measures as limits and products involving determinantal processes.

Connection between infinite measures and finite-rank perturbations of Bessel point processes.

Abstract

Infinite determinantal measures introduced in this note are inductive limits of determinantal measures on an exhausting family of subsets of the phase space. Alternatively, an infinite determinantal measure can be described as a product of a determinantal process and a convergent, but not integrable, multiplicative functional. Theorem 2, the main result announced in this note, gives an explicit description for the ergodic decomposition of infinite Pickrell measures on the spaces of infinite complex matrices in terms of infinite determinantal measures obtained by finite-rank perturbations of Bessel point processes.