Mikhail SODIN

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Part III. Related works:
T.L.Malevich (1973): non-trivial lower bound for the mean number of
components. She considered C 2 -smooth Gaussian functions F on R 2 with
positive covariance function and proved that 0 < c ≤ EN(R; F )/R 2 ≤ C < ∞.
Her proof uses the positivity property of the covariance function, which in
many instances does not hold.
E.Bogomolny and C.Schmit (2002): bond percolation model for description of
the zero set of translation-invariant Gaussian function F on R 2 whose spectral
measure is a Lebesgue measure on the unit circumference. Their model ignores
slow decaying correlations between values of F at different points, and is far
from being rigorous. Computations based on that model are well supported by
numerics.
It is not clear whether their approach can be extended to other spectral
measures or to higher dim.
Challenge: “hidden universality law” that provides the rigorous foundation for
the work done by Bogomolny and Schmit.
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