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Bulk scaling limits for random normal matrix ensembles near singularities

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dc.contributor.advisor김판기-
dc.contributor.author서성미-
dc.date.accessioned2017-10-27T17:13:27Z-
dc.date.available2017-10-27T17:13:27Z-
dc.date.issued2017-08-
dc.identifier.other000000146333-
dc.identifier.urihttps://hdl.handle.net/10371/137159-
dc.description학위논문 (박사)-- 서울대학교 대학원 자연과학대학 수리과학부, 2017. 8. 김판기.-
dc.description.abstract본 학위 논문에서는 랜덤 정규 행렬(random normal matrix)의 고유값들이 특이점 근방에서 이루는 확률분포를 연구한다. 랜덤 정규 행렬의 고유값들은 외부 포텐셜(external potential)이 주어져있는 볼츠만-깁스(Boltzmann-Gibbs)분포를 따른다. 외부 포텐셜이 무한대 근처에서 충분히 빠르게 증가하도록 주어지면, 행렬의 크기가 커짐에 따라 고유값들은 근사적으로 평형 측도(equilibrium measure)를 따라 분포하며 복소 평면 위의 옹골집합(compact set)에 모이게 된다.

이 옹골집합 내부에서 평형 측도의 밀도함수가 0이 되는 점을 내부 특이점(bulk singularity)이라 하며, 옹골집합 내부에서 로그 특이성을 갖는 점을 원뿔 특이점(conical singularity)이라 한다. 본 학위 논문에서는 이 두 종류의 특이점 근방에서 표준화된 고유값 분포의 극한과 그 극한의 보편성(universality)에 관해 논의한다.		-
dc.description.tableofcontents1 Introduction 1
1.1 Random normal matrix models . . . . . . . . . . . . . . . . .1
1.2 Microscopic properties of random normal matrix ensembles .3
1.3 Rescaled eigenvalue system near a singularity . . . . . . . . .4

2 Preliminaries 7
2.1 Random normal matrix ensembles . . . . . . . . . . . . . . .7
2.1.1 The joint distribution of eigenvalues . . . . . . . . . .8
2.1.2 Physical context . . . . . . . . . . . . . . . . . . . . .9
2.2 Determinantal point processes . . . . . . . . . . . . . . . . . .10
2.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . .10
2.2.2 Eigenvalue processes of random normal matrices . . .11
2.3 Logarithmic potential theory . . . . . . . . . . . . . . . . . .12
2.3.1 Equilibrium measure and droplet . . . . . . . . . . . .13
2.3.2 Convergence of marginal probability measures . . . . .14
2.4 Wards identities . . . . . . . . . . . . . . . . . . . . . . . . .15

3 Rescaled point processes near a bulk singularity 18
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
3.1.1 Microscopic scale . . . . . . . . . . . . . . . . . . . . .19
3.1.2 Rescaled point processes . . . . . . . . . . . . . . . . .20
3.1.3 Canonical decomposition . . . . . . . . . . . . . . . .20
3.1.4 Example : The Mittag-Leffler ensembles . . . . . . . .21
3.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
3.3 Existence of limiting kernels . . . . . . . . . . . . . . . . . . .25
3.3.1 Local uniform boundedness of the rescaled kernel . . .26
3.3.2 Structure of limiting kernels . . . . . . . . . . . . . . .27
3.4 Properties of limiting holomorphic kernels . . . . . . . . . . .29
3.4.1 Positive matrices and reproducing kernels . . . . . . .29
3.4.2 The positivity theorem . . . . . . . . . . . . . . . . . .31
3.5 Wards equation and zero-one law . . . . . . . . . . . . . . . .31
3.5.1 The rescaled Wards equation . . . . . . . . . . . . . .32
3.5.2 The proof of Theorem 3.2.2 and Theorem 3.2.3 . . . .34
3.6 Dominant radial singularities . . . . . . . . . . . . . . . . . .38
3.7 Homogeneous singularities . . . . . . . . . . . . . . . . . . . .43

4 Conical singularities 45
4.1 Introduction and results . . . . . . . . . . . . . . . . . . . . .45
4.1.1 Perturbation of potentials . . . . . . . . . . . . . . . .45
4.1.2 Conical singularities . . . . . . . . . . . . . . . . . . .46
4.1.3 Microscopic scale . . . . . . . . . . . . . . . . . . . . .47
4.1.4 Main results . . . . . . . . . . . . . . . . . . . . . . . .48
4.1.5 Example : Mittag-Leffler ensembles. . . . . . . . . . .50
4.2 Existence of limiting kernels . . . . . . . . . . . . . . . . . . .50
4.2.1 Estimates for the reproducing kernels . . . . . . . . .51
4.2.2 Local uniform boundedness of the rescaled kernel . . .54
4.2.3 Positivity . . . . . . . . . . . . . . . . . . . . . . . . .57
4.3 Homogeneous singularities . . . . . . . . . . . . . . . . . . . .58
4.4 Johanssons marginal measure theorem . . . . . . . . . . . . .59

5 Asymptotics for the one point functions 67
5.1 Bulk singularities . . . . . . . . . . . . . . . . . . . . . . . . .67
5.1.1 Asymptotics for L0(z, z) . . . . . . . . . . . . . . . . .68
5.1.2 Asymptotics for L(z, z) . . . . . . . . . . . . . . . . .73
5.2 Conical singularities . . . . . . . . . . . . . . . . . . . . . . .77		-
dc.formatapplication/pdf-
dc.format.extent3424470 bytes-
dc.format.mediumapplication/pdf-
dc.language.isoen-
dc.publisher서울대학교 대학원-
dc.subjectRandom normal matrix-
dc.subjectBulk singularity-
dc.subjectConical singularity-
dc.subjectWard's equation-
dc.subjectUniversality-
dc.subject.ddc510-
dc.titleBulk scaling limits for random normal matrix ensembles near singularities-
dc.typeThesis-
dc.description.degreeDoctor-
dc.contributor.affiliation자연과학대학 수리과학부-
dc.date.awarded2017-08-
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