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dc.contributor.advisorYau, Horng-Tzer
dc.contributor.authorLopatto, Patrick
dc.date.accessioned2020-10-16T14:09:07Z
dc.date.created2020-05
dc.date.issued2020-04-29
dc.date.submitted2020
dc.identifier.citationLopatto, Patrick. 2020. Universality of Lévy Matrices. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences.
dc.identifier.urihttps://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37365818*
dc.description.abstractMotivated by conjectures from physics, we study the eigenvalues and eigenvectors of Lévy matrices, which are symmetric random matrices whose upper triangular entries are independent, identically distributed α-stable distributions. For α < 2, these distributions are heavy-tailed, with infinite second moment. For α ∈ (1, 2), we show that at all finite non- zero energies, Lévy matrices exhibit completely delocalized eigenvectors and local eigenvalue statistics that asymptotically match those of the Gaussian Orthogonal Ensemble. For almost all α ∈ (0, 2), we prove the same result for small energies, including zero. Additionally, for almost all α ∈ (0, 2), we analyze the statistics of eigenvector entries of Lévy matrices at small energies and show that the limiting distribution of any such entry is non-Gaussian. For entries of the eigenvector corresponding to the median eigenvalue, we identify this distribution explicitly. We also demonstrate the presence of non-trivial correlations between eigenvector entries corresponding to nearby eigenvalues. These findings contrast sharply with the known eigenvector behavior for other random matrix ensembles. Further, our results for both eigenvalues and eigenvectors generalize to a large class of heavy-tailed random matrices.
dc.description.sponsorshipMathematics
dc.format.mimetypeapplication/pdf
dc.language.isoen
dash.licenseLAA
dc.subjectrandom matrix theory
dc.subjectprobability
dc.titleUniversality of Lévy Matrices
dc.typeThesis or Dissertation
dash.depositing.authorLopatto, Patrick
dc.date.available2020-10-16T14:09:07Z
thesis.degree.date2020
thesis.degree.grantorGraduate School of Arts & Sciences
thesis.degree.grantorGraduate School of Arts & Sciences
thesis.degree.levelDoctoral
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy
thesis.degree.nameDoctor of Philosophy
dc.contributor.committeeMemberBrennecke, Christian
dc.contributor.committeeMemberLemm, Marius
dc.type.materialtext
thesis.degree.departmentMathematics
thesis.degree.departmentMathematics
dash.identifier.vireo
dash.author.emailpatricklopatto@gmail.com


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