Complex determinantal processes and \(H^1\) noise. (English) Zbl 1127.60048

Rider, Brian; Virag, Balint

Electron. J. Probab. 12, 1238-1257 (2007).

Summary: For the plane, sphere, and hyperbolic plane we consider the canonical invariant determinantal point processes \({\mathcal Z}_p\) 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 a Gaussian with mean zero and variance \(\frac{1}{2\pi}\|f\|^2_{H^1}\).

MSC:

60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
60D05	Geometric probability and stochastic geometry
30F99	Riemann surfaces

Keywords:

random matrices; invariant point process; noise limit