dc.contributor.author
dc.contributor.author
Rivera, Alejandro
dc.contributor.author
Vanneuville, Hugo
dc.contributor.author
Köhler-Schindler, Laurin
dc.date.accessioned
2021-01-22T12:58:18Z
dc.date.available
2021-01-22T08:01:29Z
dc.date.available
2021-01-22T12:58:18Z
dc.date.issued
2020-10-22
dc.identifier.uri
http://hdl.handle.net/20.500.11850/464625
dc.description.abstract
We develop techniques to study the phase transition for planar Gaussian percolation models that are not (necessarily) positively correlated. These models lack the property of positive associations (also known as the FKG inequality'), and hence many classical arguments in percolation theory do not apply. More precisely, we consider a smooth stationary centred planar Gaussian field f and, given a level ℓ∈R, we study the connectivity properties of the excursion set {f≥−ℓ}. We prove the existence of a phase transition at the critical level ℓcrit=0 under only symmetry and (very mild) correlation decay assumptions, which are satisfied by the random plane wave for instance. As a consequence, all non-zero level lines are bounded almost surely, although our result does not settle the boundedness of zero level lines (no percolation at criticality'). To show our main result: (i) we prove a general sharp threshold criterion, inspired by works of Chatterjee, that states that `sharp thresholds are equivalent to the delocalisation of the threshold location'; (ii) we prove threshold delocalisation for crossing events at large scales -- at this step we obtain a sharp threshold result but without being able to locate the threshold -- and (iii) to identify the threshold, we adapt Tassion's RSW theory replacing the FKG inequality by a sprinkling procedure. Although some arguments are specific to the Gaussian setting, many steps are very general and we hope that our techniques may be adapted to analyse other models without FKG.
en_US
dc.language.iso
en
en_US
dc.publisher
Cornell University
en_US
dc.title
The phase transition for planar Gaussian percolation models without FKG
en_US
dc.type
Working Paper
ethz.journal.title
arXiv
ethz.pages.start
2010.11770v1
en_US
ethz.size
40 p.
en_US
ethz.identifier.arxiv
2010.11770
ethz.publication.place
Ithaca, NY
en_US
ethz.publication.status
published
en_US
ethz.leitzahl
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02003 - Mathematik Selbständige Professuren::09453 - Werner, Wendelin / Werner, Wendelin
en_US
ethz.leitzahl.certified
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02003 - Mathematik Selbständige Professuren::09453 - Werner, Wendelin / Werner, Wendelin
en_US
ethz.date.deposited
2021-01-22T08:01:37Z
ethz.source
FORM
ethz.eth
yes
en_US
ethz.availability
en_US
ethz.rosetta.installDate
2021-01-22T12:58:28Z
ethz.rosetta.lastUpdated
2021-01-22T12:58:28Z
ethz.rosetta.versionExported
true
ethz.COinS