On Number Rigidity for Pfaffian Point Processes

TL;DR

This paper proves number rigidity for certain Pfaffian point processes, including Bessel and sine processes, using variance estimates and spectral measure asymptotics, advancing understanding of their structural properties.

Contribution

It establishes number rigidity for orthogonal and symplectic Bessel processes and provides a general sufficient condition for rigidity in stationary Pfaffian processes.

Findings

Orthogonal and symplectic Bessel processes are rigid.

A spectral measure asymptotics criterion for rigidity is proposed.

The results extend rigidity understanding to Pfaffian sine-processes.

Abstract

Our first result states that the orthogonal and symplectic Bessel processes are rigid in the sense of Ghosh and Peres. Our argument in the Bessel case proceeds by an estimate of the variance of additive statistics in the spirit of Ghosh and Peres. Second, a sufficient condition for number rigidity of stationary Pfaffian processes, relying on the Kolmogorov criterion for interpolation of stationary processes and applicable, in particular, to pfaffian sine-processes, is given in terms of the asymptotics of the spectral measure for additive statistics.