Rigidity of Determinantal Point Processes with the Airy, the Bessel and the Gamma Kernel

TL;DR

This paper proves that determinantal point processes with Airy, Bessel, and Gamma kernels are rigid, meaning the number of particles in a bounded domain is almost surely determined by the outside configuration.

Contribution

It establishes the rigidity of these specific determinantal point processes, extending previous methods to new kernel types.

Findings

Airy kernel process is rigid.

Bessel kernel process is rigid.

Gamma kernel process is rigid.

Abstract

A point process is said to be rigid if for any bounded domain in the phase space, the number of particles in the domain is almost surely determined by the restriction of the configuration to the complement of our bounded domain. The main result of this paper is that determinantal point processes with the Airy, the Bessel and the Gamma kernels are rigid. The proof follows the scheme of Ghosh [6], Ghosh and Peres [7]: the main step is the construction of a sequence of additive statistics with variance going to zero.