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SUMMARY:Fluctuations in the number of level set components of planar Gauss
 ian fields - Stephen Muirhead (Queen Mary)
DTSTART:20200121T140000Z
DTEND:20200121T150000Z
UID:TALK138085@talks.cam.ac.uk
CONTACT:Perla Sousi
DESCRIPTION:Gaussian fields are a model of spatial noise\, and in many app
 lications it is useful to understand the geometric structure of their leve
 l sets. There is a natural classification of geometric functionals of the 
 level sets as either `local' (e.g. length of the level sets\, volume of th
 e excursion sets\, Euler characteristic of the excursion sets) or 'non-loc
 al' (e.g. number of components of the level/excursion sets\, percolation o
 f the level/excursion sets) depending on whether there exists an integral 
 representation for the functional. In the case of `local' functionals\, fi
 rst order properties (e.g. asymptotics for the mean) are easily derived fr
 om the Kac-Rice formula\, and second order properties (e.g. asymptotics fo
 r the variance\, central limit theorems) can also be established via Weine
 r chaos expansions (Kratz--Leon '11\, Estrade--Leon '16\, Marinucci--Rossi
 --Wigman '17\, Nourdin--Peccati--Rossi '17 etc). For the `non-local' numbe
 r of level/excursion sets the analysis is more challenging\, and while fir
 st order properties were established 10 years ago by Nazarov--Sodin using 
 ergodic theoretical techniques\, up until now there have been essentially 
 no second order results. In this talk I will discuss some first steps in t
 his directions\, namely proving lower bounds on the variance which are of 
 `correct' order. Joint work with Dmitry Belyaev and Michael McAuley (Unive
 rsity of Oxford).
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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