Random Walks on Reductive Groups

Provides a self-contained introduction to the products of independent identically distributed random matrices and to their Lyapunov exponents

Explains the relevance of the theory of reductive algebraic groups and the theory of bounded operators in Banach spaces to the study of random matrices

Contains a proof of the Local Limit Theorem for the norm of the products of independent identically distributed random matrices

Texte du rabat

The classical theory of random walks describes the asymptotic behavior of sums of independent identically distributed random real variables. This book explains the generalization of this theory to products of independent identically distributed random matrices with real coefficients. Under the assumption that the action of the matrices is semisimple or, equivalently, that the Zariski closure of the group generated by these matrices is reductive - and under suitable moment assumptions, it is shown that the norm of the products of such random matrices satisfies a number of classical probabilistic laws. This book includes necessary background on the theory of reductive algebraic groups, probability theory and operator theory, thereby providing a modern introduction to the topic.

Résumé
"Benoist and Quint have written an excellent text, one that will surely become a standard reference to introduce students to the fascinating nonabelian extension of the now-classical study of random walks. ... I congratulate the authors on their well-written and timely offering, and strongly recommend that libraries order a copy of this excellent text!" (Tushar Das, MAA Reviews, November, 2017)

"This book is an exposition of the tools and perspectives needed to reach the current frontier of research in the field of random matrix products... This is a technical subject, drawing on tools from a diverse range of topics... The authors take time to explain everything at a reasonable pace." (Radhakrishnan Nair, Mathematical Reviews)

"This reviewer does not hesitate to consider this book as exceptional." (Marius Iosifescu, zbMATH, 1366.60002)

Contenu
Introduction.- Part I The Law of Large Numbers.- Stationary measures.- The Law of Large Numbers.- Linear random walks.- Finite index subsemigroups.- Part II Reductive groups.- Loxodromic elements.- The Jordan projection of semigroups.- Reductive groups and their representations.- Zariski dense subsemigroups.- Random walks on reductive groups.- Part III The Central Limit Theorem.- Transfer operators over contracting actions.- Limit laws for cocycles.- Limit laws for products of random matrices.- Regularity of the stationary measure.- Part IV The Local Limit Theorem.- The Spectrum of the complex transfer operator.- The Local limit theorem for cocycles.- The local limit theorem for products of random matrices.- Part V Appendix.- Convergence of sequences of random variables.- The essential spectrum of bounded operators.- Bibliographical comments.