CONTENTS xi 5.2.1 Algebraic noncommutative probability spaces and laws 328 5.2.2 C*-probability spaces and the weak-*topology 332 5.2.3 W*-probability spaces 341 5.3 Free independence 351 5.3.1 Independence and free independence 351 5.3.2 Free independence and combinatorics 356 5.3.3 Consequence of free independence:free convolution 362 5.3.4 Free central limit theorem 371 5.3.5 Freeness for unbounded variables 372 5.4 Link with random matrices 377 5.5 Convergence of the operator norm of polynomials of inde- pendent GUE matrices 396 5.6 Bibliographical Notes 412 Appendices 417 A Linear algebra preliminaries 417 A.1 Identities and bounds 417 A.2 Perturbations for normal and Hermitian matrices 418 A.3 Noncommutative Matrix LP-norms 419 A.4 Brief review of resultants and discriminants 420 B Topological Preliminaries 421 B.1 Generalities 421 B.2 Topological Vector Spaces and Weak Topologies 424 B.3 Banach and Polish Spaces 425 B.4 Some elements of analysis 426 Probability measures on Polish spaces 427 C.1 Generalities 427 C.2 Weak Topology 429 D Basic notions of large deviations 431 E The skew field H of quaternions,and matrix theory over F 434 E.1 Matrix terminology over F,and factorization theorems435
CONTENTS xi 5.2.1 Algebraic noncommutative probability spaces and laws 328 5.2.2 C ∗ - probability spaces and the weak-* topology 332 5.2.3 W∗ - probability spaces 341 5.3 Free independence 351 5.3.1 Independence and free independence 351 5.3.2 Free independence and combinatorics 356 5.3.3 Consequence of free independence: free convolution 362 5.3.4 Free central limit theorem 371 5.3.5 Freeness for unbounded variables 372 5.4 Link with random matrices 377 5.5 Convergence of the operator norm of polynomials of independent GUE matrices 396 5.6 Bibliographical Notes 412 Appendices 417 A Linear algebra preliminaries 417 A.1 Identities and bounds 417 A.2 Perturbations for normal and Hermitian matrices 418 A.3 Noncommutative Matrix L p -norms 419 A.4 Brief review of resultants and discriminants 420 B Topological Preliminaries 421 B.1 Generalities 421 B.2 Topological Vector Spaces and Weak Topologies 424 B.3 Banach and Polish Spaces 425 B.4 Some elements of analysis 426 C Probability measures on Polish spaces 427 C.1 Generalities 427 C.2 Weak Topology 429 D Basic notions of large deviations 431 E The skew field H of quaternions, and matrix theory over F 434 E.1 Matrix terminology over F, and factorization theorems 435
xii CONTENTS E.2 The spectral theorem and key corollaries 437 E.3 A specialized result on projectors 438 E.4 Algebra for curvature computations 439 Manifolds 441 F1 Manifolds embedded in Euclidean space 442 F.2 Proof of the coarea formula 446 F.3 Metrics,connections,curvature,hessians,and the Laplace-Beltrami operator 449 G Appendix on Operator Algebras 454 G.1 Basic definitions 454 G.2 Spectral properties 456 G.3 States and positivity 458 G.4 von Neumann algebras 459 G.5 Noncommutative functional calculus 461 H Stochastic calculus notions 463 References 468 General Conventions 484
xii CONTENTS E.2 The spectral theorem and key corollaries 437 E.3 A specialized result on projectors 438 E.4 Algebra for curvature computations 439 F Manifolds 441 F.1 Manifolds embedded in Euclidean space 442 F.2 Proof of the coarea formula 446 F.3 Metrics, connections, curvature, hessians, and the Laplace-Beltrami operator 449 G Appendix on Operator Algebras 454 G.1 Basic definitions 454 G.2 Spectral properties 456 G.3 States and positivity 458 G.4 von Neumann algebras 459 G.5 Noncommutative functional calculus 461 H Stochastic calculus notions 463 References 468 General Conventions 484
Preface The study of random matrices,and in particular the properties of their eigenval- ues,has emerged from the applications,first in data analysis and later as statisti- cal models for heavy nuclei atoms.Thus,the field of random matrices owes its existence to applications.Over the years,however,it became clear that models related to random matrices play an important role in areas of pure mathematics. Moreover,the tools used in the study of random matrices came themselves from different and seemingly unrelated branches of mathematics. At this point in time,the topic has evolved enough that the newcomer,especially if coming from the field of probability theory,faces a formidable and somewhat confusing task in trying to access the research literature.Furthermore,the back- ground expected of such a newcomer is diverse,and often has to be supplemented before a serious study of random matrices can begin. We believe that many parts of the field of random matrices are now developed enough to enable one to expose the basic ideas in a systematic and coherent way. Indeed,such a treatise,geared toward theoretical physicists,has existed for some time,in the form of Mehta's superb book [Meh91].Our goal in writing this book has been to present a rigorous introduction to the basic theory of random matri- ces,including free probability,that is sufficiently self contained to be accessible to graduate students in mathematics or related sciences,who have mastered probabil- ity theory at the graduate level,but have not necessarily been exposed to advanced notions of functional analysis,algebra or geometry.Along the way,enough tech- niques are introduced that hopefully will allow readers to continue their journey into the current research literature. This project started as notes for a class on random matrices that two of us(G.A. and O.Z.)taught in the University of Minnesota in the fall of 2003,and notes for a course in the probability summer school in St.Flour taught by A.G.in the xiii
Preface The study of random matrices, and in particular the properties of their eigenvalues, has emerged from the applications, first in data analysis and later as statistical models for heavy nuclei atoms. Thus, the field of random matrices owes its existence to applications. Over the years, however, it became clear that models related to random matrices play an important role in areas of pure mathematics. Moreover, the tools used in the study of random matrices came themselves from different and seemingly unrelated branches of mathematics. At this point in time, the topic has evolved enough that the newcomer, especially if coming from the field of probability theory, faces a formidable and somewhat confusing task in trying to access the research literature. Furthermore, the background expected of such a newcomer is diverse, and often has to be supplemented before a serious study of random matrices can begin. We believe that many parts of the field of random matrices are now developed enough to enable one to expose the basic ideas in a systematic and coherent way. Indeed, such a treatise, geared toward theoretical physicists, has existed for some time, in the form of Mehta’s superb book [Meh91]. Our goal in writing this book has been to present a rigorous introduction to the basic theory of random matrices, including free probability, that is sufficiently self contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Along the way, enough techniques are introduced that hopefully will allow readers to continue their journey into the current research literature. This project started as notes for a class on random matrices that two of us (G. A. and O. Z.) taught in the University of Minnesota in the fall of 2003, and notes for a course in the probability summer school in St. Flour taught by A. G. in the xiii
XIV PREFACE summer of 2006.The comments of participants in these courses,and in particular A.Bandyopadhyay,H.Dong,K.Hoffman-Credner,A.Klenke,D.Stanton and P.M.Zamfir,were extremely useful.As these notes evolved,we taught from them again at the University of Minnesota,the University of California at Berkeley,the Technion and Weizmann Institute,and received more much appreciated feedback from the participants in those courses.Finally,when expanding and refining these course notes,we have profited from the comments and questions of many col- leagues.We would like to thank in particular G.Ben Arous,P.Biane,P.Deift, A.Dembo,P.Diaconis,U.Haagerup,V.Jones,M.Krishnapur,Y.Peres,R.Pin- sky,G.Pisier,B.Rider,D.Shlyakhtenko,B.Solel,A.Soshnikov,R.Speicher,T. Suidan,C.Tracy,B.Virag and D.Voiculescu for their suggestions,corrections, and patience in answering our questions or explaining their work to us.Of course, any remaining mistakes and unclear passages are fully our responsibility. MINNEAPOLIS.MINNESOTA GREG ANDERSON LYON,FRANCE ALICE GUIONNET REHOVOT.ISRAEL OFER ZEITOUNI APRIL 2009
xiv PREFACE summer of 2006. The comments of participants in these courses, and in particular A. Bandyopadhyay, H. Dong, K. Hoffman-Credner, A. Klenke, D. Stanton and P.M. Zamfir, were extremely useful. As these notes evolved, we taught from them again at the University of Minnesota, the University of California at Berkeley, the Technion and Weizmann Institute, and received more much appreciated feedback from the participants in those courses. Finally, when expanding and refining these course notes, we have profited from the comments and questions of many colleagues. We would like to thank in particular G. Ben Arous, P. Biane, P. Deift, A. Dembo, P. Diaconis, U. Haagerup, V. Jones, M. Krishnapur, Y. Peres, R. Pinsky, G. Pisier, B. Rider, D. Shlyakhtenko, B. Solel, A. Soshnikov, R. Speicher, T. Suidan, C. Tracy, B. Virag and D. Voiculescu for their suggestions, corrections, and patience in answering our questions or explaining their work to us. Of course, any remaining mistakes and unclear passages are fully our responsibility. GREG ANDERSON ALICE GUIONNET OFER ZEITOUNI APRIL 2009 MINNEAPOLIS, MINNESOTA LYON, FRANCE REHOVOT, ISRAEL
1 Introduction This book is concerned with random matrices.Given the ubiquitous role that matrices play in mathematics and its application in the sciences and engineering, it seems natural that the evolution of probability theory would eventually pass through random matrices.The reality,however,has been more complicated (and interesting).Indeed,the study of random matrices,and in particular the properties of their eigenvalues,has emerged from the applications,first in data analysis(in the early days of statistical sciences,going back to Wishart [Wis281),and later as statistical models for heavy nuclei atoms,beginning with the seminal work of Wigner [Wig55].Still motivated by physical applications,at the able hands of Wigner,Dyson,Mehta and co-workers,a mathematical theory of the spectrum of random matrices began to emerge in the early 1960s,and links with various branches of mathematics,including classical analysis and number theory,were established.While much advance was initially achieved using enumerative combi- natorics,gradually,sophisticated and varied mathematical tools were introduced: Fredholm determinants(in the 1960s),diffusion processes (in the 1960s),inte- grable systems(in the 1980s and early 1990s),and the Riemann-Hilbert problem (in the 1990s)all made their appearance,as well as new tools such as the theory of free probability (in the 1990s).This wide array of tools,while attesting to the vi- tality of the field,present however several formidable obstacles to the newcomer, and even to the expert probabilist.Indeed,while much of the recent research uses sophisticated probabilistic tools,it builds on layers of common knowledge that,in the aggregate,few people possess Our goal in this book is to present a rigorous introduction to the basic theory of random matrices that would be sufficiently self contained to be accessible to grad- uate students in mathematics or related sciences,who have mastered probability theory at the graduate level,but have not necessarily been exposed to advanced notions of functional analysis,algebra or geometry.With such readers in mind,we
1 Introduction This book is concerned with random matrices. Given the ubiquitous role that matrices play in mathematics and its application in the sciences and engineering, it seems natural that the evolution of probability theory would eventually pass through random matrices. The reality, however, has been more complicated (and interesting). Indeed, the study of random matrices, and in particular the properties of their eigenvalues, has emerged from the applications, first in data analysis (in the early days of statistical sciences, going back to Wishart [Wis28]), and later as statistical models for heavy nuclei atoms, beginning with the seminal work of Wigner [Wig55]. Still motivated by physical applications, at the able hands of Wigner, Dyson, Mehta and co-workers, a mathematical theory of the spectrum of random matrices began to emerge in the early 1960s, and links with various branches of mathematics, including classical analysis and number theory, were established. While much advance was initially achieved using enumerative combinatorics, gradually, sophisticated and varied mathematical tools were introduced: Fredholm determinants (in the 1960s), diffusion processes (in the 1960s), integrable systems (in the 1980s and early 1990s), and the Riemann-Hilbert problem (in the 1990s) all made their appearance, as well as new tools such as the theory of free probability (in the 1990s). This wide array of tools, while attesting to the vitality of the field, present however several formidable obstacles to the newcomer, and even to the expert probabilist. Indeed, while much of the recent research uses sophisticated probabilistic tools, it builds on layers of common knowledge that, in the aggregate, few people possess. Our goal in this book is to present a rigorous introduction to the basic theory of random matrices that would be sufficiently self contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. With such readers in mind, we 1