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2007 Complex Determinantal Processes and $H1$ Noise
Brian Rider, Balint Virag
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Electron. J. Probab. 12: 1238-1257 (2007). DOI: 10.1214/EJP.v12-446


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$.


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Brian Rider. Balint Virag. "Complex Determinantal Processes and $H1$ Noise." Electron. J. Probab. 12 1238 - 1257, 2007.


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

zbMATH: 1127.60048
MathSciNet: MR2346510
Digital Object Identifier: 10.1214/EJP.v12-446

Primary: 60D05
Secondary: 30F99

Keywords: Determinantal process , invariant point process , noise limit , random matrices

Vol.12 • 2007
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