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hal.structure.identifier
dc.contributor.authorRhodes, Rémi
hal.structure.identifier
dc.contributor.authorVargas, Vincent
HAL ID: 739861
dc.date.accessioned2017-01-17T14:43:34Z
dc.date.available2017-01-17T14:43:34Z
dc.date.issued2015
dc.identifier.issn0926-2601
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/16167
dc.language.isoenen
dc.subjectGaussian multiplicative chaosen
dc.subjectCritical Liouville quantum gravityen
dc.subjectBrownian motionen
dc.subjectHeat kernelen
dc.subjectPotential theoryen
dc.subject.ddc519en
dc.titleLiouville Brownian motion at criticalityen
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenIn this paper, we construct the Brownian motion of Liouville Quantum Gravity with central charge c=1 (more precisely we restrict to the corresponding free field theory). Liouville quantum gravity with c=1 corresponds to two-dimensional string theory and is the conjectural scaling limit of large planar maps weighted with a O(n=2) loop model or a Q=4-state Potts model embedded in a two dimensional surface in a conformal manner. Following Garban et al. (2013), we start by constructing the critical LBM from one fixed point xR2 (or xS2), which amounts to changing the speed of a standard planar Brownian motion depending on the local behaviour of the critical Liouville measure M′(dx)=−X(x)e2X(x)dx (where X is a Gaussian Free Field, say on S2). Extending this construction simultaneously to all points in R2 requires a fine analysis of the potential properties of the measure M′. This allows us to construct a strong Markov process with continuous sample paths living on the support of M′, namely a dense set of Hausdorff dimension 0. We finally construct the Liouville semigroup, resolvent, Green function, heat kernel and Dirichlet form of (critical) Liouville quantum gravity with a c=1 central charge. In passing, we extend to quite a general setting the construction of the critical Gaussian multiplicative chaos that was initiated in Duplantier et al. (Ann. Probab. 42(5), 1769–1808, 2014), Duplantier et al. (Commun. Math. Phys. 330, 283–330 2014) and also establish new capacity estimates for the critical Gaussian multiplicative chaos.en
dc.relation.isversionofjnlnamePotential Analysis
dc.relation.isversionofjnlvol43en
dc.relation.isversionofjnlissue2en
dc.relation.isversionofjnldate2015
dc.relation.isversionofjnlpages149-197en
dc.relation.isversionofdoi10.1007/s11118-015-9467-4en
dc.identifier.urlsitehttp://hal.archives-ouvertes.fr/hal-00914387en
dc.relation.isversionofjnlpublisherKluwer Academic Publishersen
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.relation.forthcomingnonen
dc.relation.forthcomingprintnonen
dc.description.ssrncandidatenonen
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.relation.Isversionofjnlpeerreviewedouien
dc.relation.Isversionofjnlpeerreviewedouien
dc.date.updated2016-12-03T15:03:47Z
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