The Ergodic Decomposition of Infinite Pickrell Measures III. The infinite Bessel process as the limit of radial parts of finite-dimensional projections of infinite Pickrell measures

TL;DR

This paper proves that the radial parts of finite-dimensional infinite Pickrell measures converge to the infinite Bessel process, describing the ergodic decomposition and showing singularity for different parameters.

Contribution

It establishes the scaling limit of radial parts as the infinite Bessel process and characterizes the ergodic components of infinite Pickrell measures.

Findings

Radial parts converge to the infinite Bessel process

Gaussian parameter vanishes in ergodic components

Different parameter values lead to singular measures

Abstract

The third part of the paper concludes the proof of the main result --- the description of the ergodic decomposition of infinite Pickrell measures. First it is shown that the scaling limit of radial parts of finite-dimensional infinite Pickrell measures is precisely the infinite Bessel point process. It is then established that the "gaussian parameter" almost surely vanishes for our ergodic components, and the convergence to the scaling limit is then established in the space of finite measures on the space of finite measures. Finally, singularity is established for Pickrell measures corresponding to different values of the parameter.