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High Dimensional Normality of Noisy ...

Marcinek, Jake.

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High Dimensional Normality of Noisy Eigenvectors.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	High Dimensional Normality of Noisy Eigenvectors./
作者:	Marcinek, Jake.
出版者:	Ann Arbor : ProQuest Dissertations & Theses, : 2020,
面頁冊數:	145 p.
附註:	Source: Dissertations Abstracts International, Volume: 82-11, Section: B.
Contained By:	Dissertations Abstracts International82-11B.
標題:	Mathematics. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28452502
ISBN:	9798597059617

High Dimensional Normality of Noisy Eigenvectors.
Marcinek, Jake.

High Dimensional Normality of Noisy Eigenvectors. - Ann Arbor : ProQuest Dissertations & Theses, 2020 - 145 p.

Source: Dissertations Abstracts International, Volume: 82-11, Section: B.

Thesis (Ph.D.)--Harvard University, 2020.

This item must not be sold to any third party vendors.

We 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.

ISBN: 9798597059617Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Random matrix theory

High Dimensional Normality of Noisy Eigenvectors.
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300 $a 145 p.
500 $a Source: Dissertations Abstracts International, Volume: 82-11, Section: B.
500 $a Advisor: Yau, Horng-Tzer.
502 $a Thesis (Ph.D.)--Harvard University, 2020.
506 $a This item must not be sold to any third party vendors.
520 $a We 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.
590 $a School code: 0084.
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
653 $a Random matrix theory
653 $a Universality
653 $a Eigenvectors
690 $a 0405
690 $a 0642
710 2 $a Harvard University. $b Mathematics. $3 2049814
773 0 $t Dissertations Abstracts International $g 82-11B.
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