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$.
Citation
Brian Rider. Balint Virag. "Complex Determinantal Processes and $H1$ Noise." Electron. J. Probab. 12 1238 - 1257, 2007. https://doi.org/10.1214/EJP.v12-446
Information