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Spectrum of Markov generators on sparse random graphs

dc.contributor.authorChafaï, Djalil
HAL ID: 11025
ORCID: 0000-0002-1446-1428
dc.contributor.authorCaputo, Pietro
dc.contributor.authorBordenave, Charles
HAL ID: 740473
dc.date.accessioned2014-02-26T15:27:44Z
dc.date.available2014-02-26T15:27:44Z
dc.date.issued2014
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/12763
dc.language.isoenen
dc.subjectSpectral Analysisen
dc.subjectCombinatoricsen
dc.subjectFree probabilityen
dc.subjectRandom matricesen
dc.subjectRandom graphsen
dc.subject.ddc519en
dc.titleSpectrum of Markov generators on sparse random graphsen
dc.typeArticle accepté pour publication ou publié
dc.contributor.editoruniversityotherLaboratoire d'Analyse et de Mathématiques Appliquées (LAMA) http://umr-math.univ-mlv.fr/ Université Paris-Est Marne-la-Vallée (UPEMLV) – Université Paris-Est Créteil Val-de-Marne (UPEC) – CNRS : UMR8050 – Fédération de Recherche Bézout;France
dc.contributor.editoruniversityotherDipartimento di Matematica [Roma TRE] http://www.mat.uniroma3.it/ Università degli Studi Roma TRE;France
dc.contributor.editoruniversityotherInstitut de Mathématiques de Toulouse (IMT) Université Paul Sabatier (UPS) - Toulouse III – Université Toulouse le Mirail - Toulouse II – Université des Sciences Sociales - Toulouse I – Institut National des Sciences Appliquées [INSA] - Toulouse – CNRS : UMR5219;France
dc.description.abstractenWe investigate the spectrum of the infinitesimal generator of the continuous time random walk on a randomly weighted oriented graph. This is the non-Hermitian random nxn matrix L defined by L(j,k)=X(j,k) if k<>j and L(j,j)=-sum(L(j,k),k<>j), where X(j,k) are i.i.d. random weights. Under mild assumptions on the law of the weights, we establish convergence as n tends to infinity of the empirical spectral distribution of L after centering and rescaling. In particular, our assumptions include sparse random graphs such as the oriented Erdős-Rényi graph where each edge is present independently with probability p(n)->0 as long as np(n) >> (log(n))^6. The limiting distribution is characterized as an additive Gaussian deformation of the standard circular law. In free probability terms, this coincides with the Brown measure of the free sum of the circular element and a normal operator with Gaussian spectral measure. The density of the limiting distribution is analyzed using a subordination formula. Furthermore, we study the convergence of the invariant measure of L to the uniform distribution and establish estimates on the extremal eigenvalues of L.en
dc.relation.isversionofjnlnameCommunications on Pure and Applied Mathematics
dc.relation.isversionofjnlvol67en
dc.relation.isversionofjnlissue4en
dc.relation.isversionofjnldate2014
dc.relation.isversionofjnlpages621-669en
dc.relation.isversionofdoihttp://dx.doi.org/10.1002/cpa.21496en
dc.relation.isversionofjnlpublisherWiley-Blackwellen
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.relation.forthcomingnonen
dc.relation.forthcomingprintnonen
dc.description.submittednonen


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