Spectrum of Markov generators on sparse random graphs
dc.contributor.author | Chafaï, Djalil
HAL ID: 11025 ORCID: 0000-0002-1446-1428 | |
dc.contributor.author | Caputo, Pietro | |
dc.contributor.author | Bordenave, Charles
HAL ID: 740473 | |
dc.date.accessioned | 2014-02-26T15:27:44Z | |
dc.date.available | 2014-02-26T15:27:44Z | |
dc.date.issued | 2014 | |
dc.identifier.uri | https://basepub.dauphine.fr/handle/123456789/12763 | |
dc.language.iso | en | en |
dc.subject | Spectral Analysis | en |
dc.subject | Combinatorics | en |
dc.subject | Free probability | en |
dc.subject | Random matrices | en |
dc.subject | Random graphs | en |
dc.subject.ddc | 519 | en |
dc.title | Spectrum of Markov generators on sparse random graphs | en |
dc.type | Article accepté pour publication ou publié | |
dc.contributor.editoruniversityother | Laboratoire 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.editoruniversityother | Dipartimento di Matematica [Roma TRE] http://www.mat.uniroma3.it/ Università degli Studi Roma TRE;France | |
dc.contributor.editoruniversityother | Institut 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.abstracten | We 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.isversionofjnlname | Communications on Pure and Applied Mathematics | |
dc.relation.isversionofjnlvol | 67 | en |
dc.relation.isversionofjnlissue | 4 | en |
dc.relation.isversionofjnldate | 2014 | |
dc.relation.isversionofjnlpages | 621-669 | en |
dc.relation.isversionofdoi | http://dx.doi.org/10.1002/cpa.21496 | en |
dc.relation.isversionofjnlpublisher | Wiley-Blackwell | en |
dc.subject.ddclabel | Probabilités et mathématiques appliquées | en |
dc.relation.forthcoming | non | en |
dc.relation.forthcomingprint | non | en |
dc.description.submitted | non | en |