Rouhollah Ebrahimi Extreme Value Statistics of Random Normal Matrices Tese de Doutorado Thesis presented to the Programa de Pós–graduação em Física of PUC-Rio in partial fulfillment of the requirements for the degree of Doutor em Ciências – Física. Advisor : Prof. Hiroshi Nunokawa Co-advisor: Prof. Stefan Zohren Rio de Janeiro October 2017 Rouhollah Ebrahimi Extreme Value Statistics of Random Normal Matrices Thesis presented to the Programa de Pós–graduação em Física of PUC-Rio in partial fulfillment of the requirements for the degree of Doutor em Ciências – Física. Approved by the undersigned Examination Committee. Prof. Hiroshi Nunokawa Advisor Departamento de Física PUC-Rio Prof. Stefan Zohren Co-advisor University of Oxford Prof. Alexei Margalhaes Veneziani Universidade Federal do ABC Prof. Carlos Tomei Departamento de Matemática – PUC-Rio Prof. José Abdalla Helayël-Neto CBPF Prof. Domingos Humberto Urbano Marchetti USP Prof. Luis Esteban Oxman UFF Prof. Márcio da Silveira Carvalho Vice Dean of Graduate Studies Centro Técnico Científico – PUC-Rio Rio de Janeiro, October the 25th, 2017 All rights reserved. Rouhollah Ebrahimi Graduated in Physics at Shahed University in 2004 and obtained his M.Sc. degree in Physics from Sharif University of Technology in 2008. Bibliographic data Ebrahimi, Rouhollah Extreme Value Statistics of Random Normal Matrices / Rouhollah Ebrahimi; advisor: Hiroshi Nunokawa; co-advisor: Stefan Zohren. – Rio de janeiro: PUC-Rio, Departamento de Física, 2017. v., 135 f: il. color. ; 30 cm Tese (doutorado) - Pontifícia Universidade Católica do Rio de Janeiro, Departamento de Física. Inclui bibliografia 1. Física – Teses. 2. Matrizes Aleatórias Normais;. 3. Esta- tísticas de Valor Extremo;. 4. Universalidade;. 5. Polinômios Ortogonais.. I. Nunokawa, Hiroshi. II. Zohren, Stefan. III. Pontifícia Universidade Católica do Rio de Janeiro. Departa- mento de Física. IV. Título. CDD: 620.11 Dedicated to my mother and sister whose support has been very precious, and to the memory of my father. Acknowledgments I would like to thank my advisor, Hiroshi Nunokawa, for all his help and gui- dance during my Ph.D. studies. My sincere gratitude also goes to co-advisor, Stefan Zohren, who introduced me to the fascinating world of random matrices. That was a great chance for me to benefit form his deep and broad knowledge in physics and mathematics. I also appreciate his patience and remarkable character which has been an exceptional inspiration throughout my graduate life. Special thanks to Carlos Tomei, at department of mathematics at PUC-Rio, whose presence and support helped me through difficulties. Besides being a great mathematician, he has a wonderful personality. I would also like to thank the constant support of Welles Morgado and José Helayël. With respect to my visit at institute of mathematics and statistics (IME), at university of São Paulo, I am particularly grateful to Anatoli Iambartsev, who provided a pleasant environment for me to work. Financial support through CAPES, CNPq and CCPG (PUC-Rio) is kindly acknowledged. Abstract Ebrahimi, Rouhollah; Nunokawa, Hiroshi (Advisor); Zohren, Ste- fan (Co-Advisor). Extreme Value Statistics of Random Nor- mal Matrices. Rio de Janeiro, 2017. 135p. Tese de doutorado – Departamento de Física, Pontifícia Universidade Católica do Rio de Janeiro. With diverse applications in mathematics, physics, and finance, Ran- dom Matrix Theory (RMT) has recently attracted a great deal of atten- tion. While Hermitian RMT is of special importance in physics because of the Hermiticity of operators associated with observables in quantum me- chanics, non-Hermitian RMT has also attracted a considerable attention, in particular because they can be used as models for dissipative or open physical systems. However, due to the absence of a simplifying symmetry, the study of non-Hermitian random matrices is, in general, a difficult task. A special subset of non-Hermitian random matrices, the so-called random normal matrices, are interesting models to consider, since they offer more symmetry, thus making them more amenable to analytical investigations. By definition, a normal matrix M is a square matrix which commutes with y its Hermitian adjoint, i.e., [M; M ] = 0. In this thesis, we present a novel derivation of extreme value statistics (EVS) of Hermitian random matri- ces, namely the approach of orthogonal polynomials, to normal random matrices and 2D Coulomb gases in general. The strength of this approach is its physical and intuitive understanding. Firstly, this approach provides an alternative derivation of results in the literature. Precisely speaking, we show convergence of the rescaled eigenvalue with largest modulus of a Ginibre ensemble to a Gumbel distribution, as well as universality for an arbitrary radially symmetric potential which meets certain conditions. Secondly, it is shown that this approach can be generalised to obtain con- vergence of the eigenvalue with smallest modulus and its universality at the finite inner edge of the eigenvalue support. One interesting aspect of this work is the fact that we can use standard techniques from Hermitian random matrices to obtain the EVS of non-Hermitian random matrices. Keywords Random Normal Matrices; Extreme Value Statistics; Universality; Orthogonal Polynomials. Resumo Ebrahimi, Rouhollah; Nunokawa, Hiroshi; Zohren, Stefan. Esta- tísticas de Valor Extremo de Matrizes Aleatórias Normais. Rio de Janeiro, 2017. 135p. Tese de Doutorado – Departamento de Física, Pontifícia Universidade Católica do Rio de Janeiro. Com diversas aplicações em matemática, física e finanças, Teoria das Matrizes Aleatórias (RMT) recentemente atraiu muita atenção. Enquanto o RMT Hermitiano é de especial importância na física por causa da Her- menticidade de operadores associados a observáveis em mecânica quân- tica, O RMT não-Hermitiano também atraiu uma atenção considerável, em particular porque eles podem ser usados como modelos para sistemas físicos dissipativos ou abertos. No entanto, devido à ausência de uma si- metria simplificada, o estudo de matrizes aleatórias não-Hermitianas é, em geral, uma tarefa difícil. Um subconjunto especial de matrizes aleat órias não-Hermitianas, as chamadas matrizes aleat’orias normais, são modelos interessantes a serem considerados, uma vez que oferecem mais simetria, tornando-as mais acessíveis às investigções analíticas. Por definição, uma matriz normal M é uma matriz quadrada que troca com seu adjunto Hermitiano. Nesta tese, amplicamos a derivação de estatísticas de valores extremos (EVS) de matrizes aleatórias Hermitianas, com base na aborda- gem de polinômios ortogonais, em matrizes aleatórias normais e em gases Coulomb 2D em geral. A força desta abordagem a sua compreensão física e intuitiva. Em primeiro lugar, essa abordagem fornece uma derivação al- ternativa de resultados na literatura. Precisamente falando, mostramos a convergência do autovalor redimensionado com o maior módulo de um conjunto de Ginibre para uma distribuição de Gumbel, bem como a uni- versalidade para um potencial arbitrário radialmente simtérico que atenda certas condições. Em segundo lugar, mostra-se que esta abordagem pode ser generalizada para obter a convergência do autovalor com menor mó- dulo e sua universalidade no limite interno finito do suporte do autovalor. Um aspecto interessante deste trabalho é o fato de que podemos usar técnicas padrão de matrizes aleatórias Hermitianas para obter o EVS de matrizes aleatórias não Hermitianas. Palavras-chave Matrizes Aleatórias Normais; Estatísticas de Valor Extremo; Uni- versalidade; Polinômios Ortogonais. Table of contents List of figures 10 List of tables 12 1 Introduction 15 2 Basic Notions of Probability Theory and Statistics 19 2.1 Univariate Random Variables 19 2.1.1 Special Probability Distributions 22 2.1.2 Moment-Generating Functions 24 2.2 Multivariate Random Variables 26 2.3 Convergence of Sequence of Random Variables 27 2.3.1 Convergence in Probability and Weak Law of Large Numbers 27 2.3.2 Convergence in Distribution and Central Limit Theorem 29 2.4 Tail Distribution 31 2.4.1 Concentration Inequalities and Tail Bounds 31 2.4.2 Classification of tails 34 3 Asymptotic Models of Classical Extreme Value Theory 37 3.1 Model Formulation 37 3.2 The Three Types of Extreme Value Distributions 39 3.3 The Generalized Extreme Value Distribution for Maxima 40 3.4 The Generalized Extreme Value Distribution for Minima 44 3.5 Domains of Attraction 45 4 Introduction to Random Matrix Theory 49 4.1 Motivation 49 4.2 Classical Ensembles of Random Matrices 50 4.3 Universality of Random Matrix Theories 54 4.3.1 Level Spacing Distribution 55 4.3.2 Eigenvalue Distribution 57 5 Gaussian Unitary Ensemble 58 5.1 Spectral Representation of the Partition Function of GUE 58 5.2 Methods of Analysis the Large N Limit 60 5.2.1 Saddle Point Method for GUE 60 5.2.2 Resolvent Formalism for Unitary Ensembles with Arbitrary Potential 63 5.2.3 Orthogonal Polynomials Method for Unitary Ensemble with Arbitrary Potential 69 5.2.4 Results of Orthogonal Polynomials Method for GUE 72 6 Ginibre Ensemble 74 6.1 Resolvent Formalism 74 6.2 Diagrammatic Method for Hermitian Random Matrices 78 6.3 Diagrammatic Method for Non-Hermitian Random Matrices 81 7 Random Normal Matrix Ensembles 84 7.1 Eigenvalue Representation of the Partition Function 84 7.2 Saddle Point Method 85 7.3 Orthogonal Polynomials Method 89 8 Extreme Value Statistics of GUE 93 8.1 Tracy-Widom Law for the Classical Gaussian Ensembles 93 8.2 Orthogonal Polynomials Approach to the EVS