Electronic Journal of Probability

Complex Determinantal Processes and $H1$ Noise

Brian Rider and Balint Virag

Full-text: Open access

Abstract

For the plane, sphere, and hyperbolic plane we consider the canonical invariant determinantal point processes $\mathcal Z_\rho$ with intensity $\rho d\nu$, where $\nu$ is the corresponding invariant measure. We show that as $\rho\to\infty$, after centering, these processes converge to invariant $H^1$ noise. More precisely, for all functions $f\in H^1(\nu) \cap L^1(\nu)$ the distribution of $\sum_{z\in \mathcal Z} f(z)-\frac{\rho}{\pi} \int f d \nu$ converges to Gaussian with mean zero and variance $ \frac{1}{4 \pi} \|f\|_{H^1}^2$.

Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 45, 1238-1257.

Dates
Accepted: 9 October 2007
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464818516

Digital Object Identifier
doi:10.1214/EJP.v12-446

Mathematical Reviews number (MathSciNet)
MR2346510

Zentralblatt MATH identifier
1127.60048

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 30F99: None of the above, but in this section

Keywords
determinantal process random matrices invariant point process noise limit

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Rider, Brian; Virag, Balint. Complex Determinantal Processes and $H1$ Noise. Electron. J. Probab. 12 (2007), paper no. 45, 1238--1257. doi:10.1214/EJP.v12-446. https://projecteuclid.org/euclid.ejp/1464818516


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