Show simple item record

dc.contributor.advisorYau, Horng-Tzer
dc.contributor.authorMarcinek, Jake Boleslaw
dc.date.accessioned2020-10-16T13:49:48Z
dc.date.created2020-05
dc.date.issued2020-05-15
dc.date.submitted2020
dc.identifier.citationMarcinek, Jake Boleslaw. 2020. High Dimensional Normality of Noisy Eigenvectors. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences.
dc.identifier.urihttps://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37365747*
dc.description.abstractWe study joint eigenvector distributions for large symmetric matrices in the presence of weak noise. Our main result asserts that every submatrix in the orthogonal matrix of eigenvectors converges to a multidimensional Gaussian distribution. The proof involves analyzing the stochastic eigenstate equation (SEE) which describes the Lie group valued flow of eigenvectors induced by matrix valued Brownian motion. We consider the associated colored eigenvector moment flow defining an SDE on a particle configuration space. This flow extends the eigenvector moment flow to the multicolor setting. However, it is no longer driven by an underlying Markov process on configuration space due to the lack of positivity in the semigroup kernel. Nevertheless, we prove the dynamics admit sufficient averaged decay and contractive properties. This allows us to establish optimal time of relaxation to equilibrium for the colored eigenvector moment flow and prove joint asymptotic normality for eigenvectors. Applications in random matrix theory include the explicit computations of joint eigenvector distributions for general Wigner type matrices and sparse graph models when corresponding eigenvalues lie in the bulk of the spectrum, as well as joint eigenvector distributions for L\'evy matrices when the eigenvectors correspond to small energy levels.
dc.description.sponsorshipMathematics
dc.format.mimetypeapplication/pdf
dc.language.isoen
dash.licenseLAA
dc.subjectrandom matrix theory
dc.subjectuniversality
dc.subjecteigenvectors
dc.titleHigh Dimensional Normality of Noisy Eigenvectors
dc.typeThesis or Dissertation
dash.depositing.authorMarcinek, Jake Boleslaw
dc.date.available2020-10-16T13:49:48Z
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.committeeMemberLu, Yue M.
dc.contributor.committeeMemberKe, Tracy
dc.type.materialtext
thesis.degree.departmentMathematics
thesis.degree.departmentMathematics
dash.identifier.vireo
dash.author.emailmarcinek@alumni.harvard.edu


Files in this item

Thumbnail
Thumbnail
Thumbnail
Thumbnail
Thumbnail
Thumbnail
Thumbnail
Thumbnail
Thumbnail
Thumbnail
Thumbnail
Thumbnail
Thumbnail

This item appears in the following Collection(s)

Show simple item record