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Capturing Noncommutativity in Nonhomogeneous Random Matrices
Capturing Noncommutativity in Nonhomogeneous Random Matrices
상세정보
- 자료유형
- 학위논문 서양
- 최종처리일시
- 20250211151453
- ISBN
- 9798382807867
- DDC
- 519
- 서명/저자
- Capturing Noncommutativity in Nonhomogeneous Random Matrices
- 발행사항
- [Sl] : Princeton University, 2024
- 발행사항
- Ann Arbor : ProQuest Dissertations & Theses, 2024
- 형태사항
- 107 p
- 주기사항
- Source: Dissertations Abstracts International, Volume: 85-12, Section: B.
- 주기사항
- Advisor: van Handel, Ramon.
- 학위논문주기
- Thesis (Ph.D.)--Princeton University, 2024.
- 초록/해제
- 요약Random matrices are ubiquitous across many fields - physics, computer science, applied and pure mathematics. Oftentimes the random matrix of interest will have non-trivial structure - entries that are dependent and have potentially different means and variances (e.g. sparse Wigner matrices, matrices corresponding to adjacencies of random graphs, sample covariance matrices). This thesis presents novel findings concerning the spectrum of such random matrices, which we say are nonhomogeneous. In particular, we focus on random matrices X = ∑ni=1 Xi such that Xi are independent, but not necessarily identically distributed. First, we consider X with independent Gaussian entries and an arbitrary variance profile. In this setting we show that ∥X∥ exhibits superconcetration, i.e. fluctuations of ∥X∥ are of smaller scale than that predicted by classical concentration inequalities. Moreover, we derive upper tail estimates for ∥X∥, which can be viewed as an extension of Tracy-Widom asymptotics for classical ensembles. Next, we show that if we instead assume that maxi ∥Xi∥ has finite second moment, then under some fairly general conditions the spectrum of X lies close to that of a Gaussian random matrix with the same mean and covariance. Whilst the proofs behind these facts differ substantially, the key idea underlying both arguments is to take advantage of noncommutativity of the summands Xi , rather than to find a way to treat Xi as scalars, as was frequently done in earlier works on matrix concentration inequalities. As a consequence, we improve upon many of the previously known results for arbitrary X, as well as obtain novel conclusions in specialized settings, such as random graphs.
- 일반주제명
- Applied mathematics
- 일반주제명
- Mathematics
- 일반주제명
- Computational physics
- 키워드
- Random graph
- 키워드
- Nonhomogeneous
- 기타저자
- Princeton University Applied and Computational Mathematics
- 기본자료저록
- Dissertations Abstracts International. 85-12B.
- 전자적 위치 및 접속
- 로그인 후 원문을 볼 수 있습니다.
MARC
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■006m o d
■007cr#unu||||||||
■020 ▼a9798382807867
■035 ▼a(MiAaPQ)AAI31296895
■040 ▼aMiAaPQ▼cMiAaPQ
■0820 ▼a519
■1001 ▼aBrailovskaya, Tatiana.▼0(orcid)0000-0001-7514-5106
■24510▼aCapturing Noncommutativity in Nonhomogeneous Random Matrices
■260 ▼a[Sl]▼bPrinceton University▼c2024
■260 1▼aAnn Arbor▼bProQuest Dissertations & Theses▼c2024
■300 ▼a107 p
■500 ▼aSource: Dissertations Abstracts International, Volume: 85-12, Section: B.
■500 ▼aAdvisor: van Handel, Ramon.
■5021 ▼aThesis (Ph.D.)--Princeton University, 2024.
■520 ▼aRandom matrices are ubiquitous across many fields - physics, computer science, applied and pure mathematics. Oftentimes the random matrix of interest will have non-trivial structure - entries that are dependent and have potentially different means and variances (e.g. sparse Wigner matrices, matrices corresponding to adjacencies of random graphs, sample covariance matrices). This thesis presents novel findings concerning the spectrum of such random matrices, which we say are nonhomogeneous. In particular, we focus on random matrices X = ∑ni=1 Xi such that Xi are independent, but not necessarily identically distributed. First, we consider X with independent Gaussian entries and an arbitrary variance profile. In this setting we show that ∥X∥ exhibits superconcetration, i.e. fluctuations of ∥X∥ are of smaller scale than that predicted by classical concentration inequalities. Moreover, we derive upper tail estimates for ∥X∥, which can be viewed as an extension of Tracy-Widom asymptotics for classical ensembles. Next, we show that if we instead assume that maxi ∥Xi∥ has finite second moment, then under some fairly general conditions the spectrum of X lies close to that of a Gaussian random matrix with the same mean and covariance. Whilst the proofs behind these facts differ substantially, the key idea underlying both arguments is to take advantage of noncommutativity of the summands Xi , rather than to find a way to treat Xi as scalars, as was frequently done in earlier works on matrix concentration inequalities. As a consequence, we improve upon many of the previously known results for arbitrary X, as well as obtain novel conclusions in specialized settings, such as random graphs.
■590 ▼aSchool code: 0181.
■650 4▼aApplied mathematics
■650 4▼aMathematics
■650 4▼aComputational physics
■653 ▼aRandom matrix theory
■653 ▼aRandom graph
■653 ▼aCovariance matrices
■653 ▼aNonhomogeneous
■690 ▼a0364
■690 ▼a0405
■690 ▼a0216
■71020▼aPrinceton University▼bApplied and Computational Mathematics.
■7730 ▼tDissertations Abstracts International▼g85-12B.
■790 ▼a0181
■791 ▼aPh.D.
■792 ▼a2024
■793 ▼aEnglish
■85640▼uhttp://www.riss.kr/pdu/ddodLink.do?id=T17161847▼nKERIS▼z이 자료의 원문은 한국교육학술정보원에서 제공합니다.
