Number rigid determinantal point processes induced by generalized Cantor sets

TL;DR

This paper demonstrates that for any measure between 0 and 1, there exists a generalized Cantor set inducing a translation-invariant determinantal point process on the real line that exhibits Ghosh-Peres number rigidity, expanding understanding of rigidity phenomena.

Contribution

It constructs a family of generalized Cantor sets with prescribed measure that induce number rigid determinantal point processes, revealing new instances of rigidity.

Findings

Existence of generalized Cantor sets with any measure in (0,1)

Corresponding determinantal processes are Ghosh-Peres number rigid

Extends rigidity results to new classes of fractal-induced processes

Abstract

We consider the Ghosh-Peres number rigidity of translation-invariant determinantal point processes on the real line $\mathbb{R}$, whose correlation kernels are induced by the Fourier transform of the indicators of generalized Cantor sets in the unit interval. Our main results show that for any given $\theta\in(0,1)$, there exists a generalized Cantor set with Lebesgue measure $\theta$, such that the corresponding determinantal point process is Ghosh-Peres number rigid.