Band-limited mimicry of point processes by point processes supported on a lattice

Abstract

We say that one point process on the line $\mathbb{R}$ mimics another at a bandwidth $B$ if for each $n\ge 1$ the two point processes have $n$-level correlation functions that agree when integrated against all bandlimited test functions on bandwidth $[-B,B]$. This paper asks the question of for what values $a$ and $B$ can a given point process on the real line be mimicked at bandwidth $B$ by a point process supported on the lattice $a\mathbb{Z}$. For Poisson point processes we give a complete answer for allowed parameter ranges $(a,B)$, and for the sine process we give existence and nonexistence regions for parameter ranges. The results for the sine process have an application to the Alternative Hypothesis regarding the scaled spacing of zeros of the Riemann zeta function, given in a companion paper.

25 pages, 1 figure. Companion paper to arXiv:1905.12123. Incorporates referee suggestions

References in corpus (8)

• Maximum of the characteristic polynomial of random unitary matrices
• On the maximum of the C$\beta$E field
• Realizability of point processes
• From Sine kernel to Poisson statistics
• The sine process under the influence of a varying potential
• Translation invariant realizability problem on the $d-$dimensional lattice: an explicit construction
• On the critical points of random matrix characteristic polynomials and of the Riemann $\xi$-function
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Cited by in corpus (1)

• Autocorrelations of characteristic polynomials for the Alternative Circular Unitary Ensemble