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Spectral Statistics of Non-Hermitian and Non-Linear Random Matrix Models
Spectral Statistics of Non-Hermitian and Non-Linear Random Matrix Models
상세정보
- 자료유형
- 학위논문 서양
- 최종처리일시
- 20260202103527
- ISBN
- 9798280712263
- DDC
- 510
- 저자명
- Dubova, Sofiia.
- 서명/저자
- Spectral Statistics of Non-Hermitian and Non-Linear Random Matrix Models
- 발행사항
- [Sl] : Harvard University, 2025
- 발행사항
- Ann Arbor : ProQuest Dissertations & Theses, 2025
- 형태사항
- 275 p
- 주기사항
- Source: Dissertations Abstracts International, Volume: 86-12, Section: A.
- 주기사항
- Advisor: Yau, Horng-Tzer.
- 학위논문주기
- Thesis (Ph.D.)--Harvard University, 2025.
- 초록/해제
- 요약This thesis presents several results on the universality phenomena of large random matrices. The first part of this thesis concerns the ensemble of non-Hermitian random matrices with independent identically distributed (i.i.d.) entries. We study the behavior of the eigenvalues and eigenvectors of this ensemble in two symmetry classes: with real-valued and complex-valued entries.Firstly, we consider an ensemble of real non-symmetric i.i.d. matrices with entries that have finite moments. We show that its k-point correlation function in the bulk away from the real line converges to a universal limit. This work builds on the previous result of Maltsev and Osman, showing the local bulk universality in the complex for the complex version of this ensemble. Secondly, we consider a constant-size subset of left and right eigenvectors of an N x N i.i.d. complex non-Hermitian matrix associated with the eigenvalues with pairwise distances at least N−1/2 +ε . We show that arbitrary constant rank projections of these eigenvectors are asymptotically Gaussian and jointly independent.Finally, motivated by applications in statistics, we consider certain large random matrices, called random inner-product kernel matrices, which are essentially given by a non-linear function f applied entrywise to a sample-covariance matrix, f(XT X), where X ∈ RdxN is random and normalized in such a way that f typically has order-one arguments. We consider the polynomial regime, where N ≍ dℓ for some ℓ 0. Earlier work by various authors showed that, when the columns of X are either uniform on the sphere or standard Gaussian vectors, and when ℓ is an integer (the linear regime ℓ = 1 is particularly well-studied), the bulk eigenvalues of such matrices behave in a simple way: They are asymptotically given by the free convolution of the semicircular and Marcenko-Pastur distributions, with relative weights given by expanding f in the Hermite basis. In the final part of this thesis, we show that this phenomenon is universal, holding as soon as X has i.i.d. entries with all finite moments.
- 일반주제명
- Mathematics
- 일반주제명
- Mathematics education
- 일반주제명
- Statistics
- 기타저자
- Harvard University Mathematics
- 기본자료저록
- Dissertations Abstracts International. 86-12A.
- 전자적 위치 및 접속
- 로그인 후 원문을 볼 수 있습니다.
MARC
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■006m o d
■007cr#unu||||||||
■020 ▼a9798280712263
■035 ▼a(MiAaPQ)AAI32039442
■040 ▼aMiAaPQ▼cMiAaPQ
■0820 ▼a510
■1001 ▼aDubova, Sofiia.▼0(orcid)0009-0009-9392-6489
■24510▼aSpectral Statistics of Non-Hermitian and Non-Linear Random Matrix Models
■260 ▼a[Sl]▼bHarvard University▼c2025
■260 1▼aAnn Arbor▼bProQuest Dissertations & Theses▼c2025
■300 ▼a275 p
■500 ▼aSource: Dissertations Abstracts International, Volume: 86-12, Section: A.
■500 ▼aAdvisor: Yau, Horng-Tzer.
■5021 ▼aThesis (Ph.D.)--Harvard University, 2025.
■520 ▼aThis thesis presents several results on the universality phenomena of large random matrices. The first part of this thesis concerns the ensemble of non-Hermitian random matrices with independent identically distributed (i.i.d.) entries. We study the behavior of the eigenvalues and eigenvectors of this ensemble in two symmetry classes: with real-valued and complex-valued entries.Firstly, we consider an ensemble of real non-symmetric i.i.d. matrices with entries that have finite moments. We show that its k-point correlation function in the bulk away from the real line converges to a universal limit. This work builds on the previous result of Maltsev and Osman, showing the local bulk universality in the complex for the complex version of this ensemble. Secondly, we consider a constant-size subset of left and right eigenvectors of an N x N i.i.d. complex non-Hermitian matrix associated with the eigenvalues with pairwise distances at least N−1/2 +ε . We show that arbitrary constant rank projections of these eigenvectors are asymptotically Gaussian and jointly independent.Finally, motivated by applications in statistics, we consider certain large random matrices, called random inner-product kernel matrices, which are essentially given by a non-linear function f applied entrywise to a sample-covariance matrix, f(XT X), where X ∈ RdxN is random and normalized in such a way that f typically has order-one arguments. We consider the polynomial regime, where N ≍ dℓ for some ℓ 0. Earlier work by various authors showed that, when the columns of X are either uniform on the sphere or standard Gaussian vectors, and when ℓ is an integer (the linear regime ℓ = 1 is particularly well-studied), the bulk eigenvalues of such matrices behave in a simple way: They are asymptotically given by the free convolution of the semicircular and Marcenko-Pastur distributions, with relative weights given by expanding f in the Hermite basis. In the final part of this thesis, we show that this phenomenon is universal, holding as soon as X has i.i.d. entries with all finite moments.
■590 ▼aSchool code: 0084.
■650 4▼aMathematics
■650 4▼aMathematics education
■650 4▼aStatistics
■653 ▼aSample-covariance matrix
■653 ▼aPolynomial regime
■653 ▼aNon-Hermitian matrix
■690 ▼a0405
■690 ▼a0280
■690 ▼a0463
■71020▼aHarvard University▼bMathematics.
■7730 ▼tDissertations Abstracts International▼g86-12A.
■790 ▼a0084
■791 ▼aPh.D.
■792 ▼a2025
■793 ▼aEnglish
■85640▼uhttp://www.riss.kr/pdu/ddodLink.do?id=T17357546▼nKERIS▼z이 자료의 원문은 한국교육학술정보원에서 제공합니다.
