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Abstract
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For a smooth, stationary, planar Gaussian field, we consider the number of connected
components of its excursion set (or level set) contained in a large square of area
.
The mean number of components is known to be of order
for
generic fields and all levels. We show that for certain fields with positive spectral
density near the origin (including the Bargmann–Fock field), and for certain levels
,
these random variables have fluctuations of order at least
, and hence variance
of order at least
.
In particular this holds for excursion sets when
is in some neighbourhood of zero, and it holds for excursion/level sets
when
is sufficiently large. We prove stronger fluctuation lower bounds of order
for
in the case that the spectral density has a singularity at the origin.
Finally we show that the number of excursion/level sets for the
random plane wave at certain levels has fluctuations of order at least
, and hence variance
of order at least .
We expect that these bounds are of the correct order, at least for generic
levels.
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Keywords
Gaussian fields, level sets, excursion sets, nodal sets,
fluctuations, variance bounds
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Mathematical Subject Classification
Primary: 58K05, 60G15, 60G60
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Milestones
Received: 2 December 2020
Revised: 20 August 2021
Accepted: 15 September 2021
Published: 11 May 2022
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