# Fluctuations of the number of excursion sets of planar Gaussian fields

@article{Beliaev2019FluctuationsOT, title={Fluctuations of the number of excursion sets of planar Gaussian fields}, author={Dmitry Beliaev and Michael Mcauley and Stephen Muirhead}, journal={arXiv: Probability}, year={2019} }

The number of connected components of the excursion set above a level $\ell$ (or level set at $\ell$) of a smooth planar Gaussian field in the ball of radius $R$ is known to have mean of order $R^2$ for any $\ell$. We show that for certain fields with positive spectral density near the origin (including the Bargmann-Fock field), and for certain values of $\ell$, these random variables have fluctuations of order at least $R$, and hence, variance of order at least $R^2$. In particular this holds… CONTINUE READING

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