An Introduction to Random Matrices (Cambridge Studies in Advanced Mathematics, Series Number 118) by Greg W. Anderson (PDF)

Description

The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory 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. Useful background material is collected in the appendices and exercises are also included throughout to test the reader’s understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

User’s Reviews

Editorial Reviews: Review “… the authors here have done an admirable job in presenting in a coherent and self-contained fashion a significant number of “core” topics of random matrix theory… this is a very valuable new reference for the subject, incorporating many modern results and perspectives that are not present in earlier texts on this topic. This book would serve as an excellent foundation with which to begin studying other aspects of random matrix theory.” Terence Tao, Mathematical Reviews Book Description A rigorous introduction to the basic theory of random matrices designed for graduate students with a background in probability theory. About the Author Greg W. Anderson is Professor in the School of Mathematics at the University of Minnesota.Alice Guionnet is Director of Research at the Ecole Normale Supérieure in Lyon and the Centre National de la Recherche Scientifique (CNRS).Ofer Zeitouni is Professor of Mathematics at both the University of Minnesota and the Weizmann Institute of Science, Israel. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐My real rating would perhaps be a 4, but I do wish to counterbalance the other review, which seems a bit unfair.First, a review of the book would be most helpful if it evaluated the product rather than its stated goal. The intended audience is announced to be students familiar with at least graduate-level probability, but who have the persistence and mathematical sophistication to learn tools as they are needed for results. These tools are surprisingly well-summarized and give the reader some confidence that they won’t have to take specialized topics courses in each of algebra, geometry, functional analysis just to understand the proofs of an interesting theorem.Now there are a variety of writing styles in mathematics, and the popular ones are not necessarily the most useful to mathematical researchers. The style adopted here emphasizes details with the goal of developing the reader’s technique, rather than of imparting a cocktail party’s knowledge of the subject. This has been my primary complaint of most other introductions that I’ve found on the web, along with the complaint of many sources’ omitting large segments of the field and being too narrow. All three of the authors are highly respected in this area, and I suspect this book is a product of their having synthesized respective notes/monographs reflecting their taste for what is most important or interesting. The outcome is a surprisingly broad coverage that illustrates how truly colorful this area is.I really dislike the rather common trend of some mathematics texts to spend many chapters developing machinery and then prove big results with apparent effortlessness. This has always felt to me to be disingenuous, especially with regard to the original development of insights that lead to solutions of big problems or programs. It seems much better to motivate the big theorems first, then develop ancillary results and machinery (to whatever degree of preferred sophistication) as the insights from exploring the relevant objects dictate to get to a proof.Because of this preference, I was very happy to see the authors’ choice to leave the “generalities” to chapter 4 and open immediately with an “elementary” proof of the most important result of the subject, the convergence of the empirical distribution of Wigner matrix eigenvalues to the Semicircle law. Even though it has been pointed out to me that the combinatorics might have been less painful with some additional assumptions on the distributions of the entries, the combinatorial technique at this level of sophistication is used repeatedly in Chapter 2.The development from there seems natural and (on the contrary to the other review) well-organized: answer more questions about the nature of this convergence, add assumptions (namely, symmetry) to yield more detailed results, and use these as a starting point (Chapter 3) to explore local questions about the eigenvalues. Some important topics (such as the Marchenko-Pastur law) are left to the exercises, which so far seem to be well crafted in testing the reader’s grasp of the arguments. Chapter 4 then deals with some of the structures of prior results in greater generality. Perhaps Chapter 4 could be described as a bit unorganized, but the authors mention that the topics were chosen as an indication of the flavor of new techniques and directions compared to the standard material of chapters 2 and 3. It makes sense to close the book with the topic of free probability in chapter 5 since this is a rather new concept to be applied to this field relative to the other techniques.I should note that, besides indulging in some skimming of various sections, I’ve only carefully worked through early portions of chapter 2 and the latter sections of chapter 4 concerning Dyson BM and tridiagonal matrix models, so perhaps my opinion will change. But so far, my only real complaint is that some proof-sketches, intended to illuminate, require more effort and feel less natural to me than the proofs themselves (which are always admirably tight and retrospectively quite clear). This can be unfortunate because these sketches can interrupt the pace of the exposition and cause the reader to lose place. If a reader also finds these confusing, I’d recommend skipping these and returning later to appreciate the authors’ reason for including it.Overall, I am excited to continue working through the text and would recommend it to anyone who might want to work in this field, or at the very least, to anyone who needs a sufficiently detailed and broad reference for the subject.

⭐It’s random alright – the selection, organization and “exposition”all qualify as random processes. The problem is not so much thatthe book is exclusively directed to the narrow specialized worldof probabilists, it’s that the authors simply have no talent forwriting.Furthermore, by now many have jumped onto the RMT bandwagon and anynumber of surveys and even course notes are immediately availableonline free. Indeed, this tripe itself already appears on theweb and should never have been published in the first place.

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