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The heat equation shrinks Ising droplets to points

dc.contributor.authorLacoin, Hubert
dc.contributor.authorSimenhaus, François
dc.contributor.authorToninelli, Fabio Lucio
HAL ID: 17419
ORCID: 0000-0003-1710-8811
dc.date.accessioned2013-10-22T07:49:20Z
dc.date.available2013-10-22T07:49:20Z
dc.date.issued2015
dc.identifier.issn0010-3640
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/11904
dc.language.isoenen
dc.subjectAnisotropy
dc.subjectIsing model
dc.subject.ddc519en
dc.titleThe heat equation shrinks Ising droplets to points
dc.typeArticle accepté pour publication ou publié
dc.contributor.editoruniversityotherInstitut Camille Jordan (ICJ) CNRS : UMR5208 – Université Claude Bernard - Lyon I – Ecole Centrale de Lyon – Institut National des Sciences Appliquées (INSA) : - LYON – Université Jean Monnet - Saint-Etienne;France
dc.description.abstractenLet D be a bounded, smooth enough domain of R^2. For L>0 consider the continuous time, zero-temperature heat bath dynamics for the nearest-neighbor Ising model on (Z/L)^2 (the square lattice with lattice spacing 1/L) with initial condition such that \sigma_x=-1 if x\in D and \sigma_x=+ otherwise. We prove the following classical conjecture due to H. Spohn: In the diffusive limit where time is rescaled by L^2 and L tends to infinity, the boundary of the droplet of "-" spins follows a deterministic anisotropic curve-shortening flow, such that the normal velocity is given by the local curvature times an explicit function of the local slope. Locally, in a suitable reference frame, the evolution of the droplet boundary follows the one-dimensional heat equation. To our knowledge, this is the first proof of mean curvature-type droplet shrinking for a lattice model with genuine microscopic dynamics. An important ingredient is our recent work, where the case of convex D was solved. The other crucial point in the proof is obtaining precise regularity estimates on the deterministic curve shortening flow. This builds on geometric and analytic ideas of Grayson, Gage-Hamilton, Gage-Li, Chou-Zhu and others.
dc.relation.isversionofjnlnameCommunications on Pure and Applied Mathematics
dc.relation.isversionofjnlvol68
dc.relation.isversionofjnlissue9
dc.relation.isversionofjnldate2015
dc.relation.isversionofjnlpages1640-1681
dc.relation.isversionofdoihttp://dx.doi.org/10.1002/cpa.21533
dc.identifier.urlsitehttp://arxiv.org/abs/1306.4507v1
dc.relation.isversionofjnlpublisherInterscience Publishers
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.relation.forthcomingprintoui
dc.description.submittednonen
dc.description.ssrncandidatenon
dc.description.halcandidateoui
dc.description.readershiprecherche
dc.description.audienceInternational
dc.relation.Isversionofjnlpeerreviewedoui
dc.date.updated2016-09-24T15:50:21Z


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