Random Walks On Reductive Groups
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| Author | : Yves Benoist |
| Publisher | : Springer |
| Total Pages | : 319 |
| Release | : 2016-10-20 |
| Genre | : Mathematics |
| ISBN | : 3319477218 |
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.
| Author | : Steven P. Lalley |
| Publisher | : Springer Nature |
| Total Pages | : 373 |
| Release | : 2023-05-08 |
| Genre | : Mathematics |
| ISBN | : 3031256328 |
This text presents the basic theory of random walks on infinite, finitely generated groups, along with certain background material in measure-theoretic probability. The main objective is to show how structural features of a group, such as amenability/nonamenability, affect qualitative aspects of symmetric random walks on the group, such as transience/recurrence, speed, entropy, and existence or nonexistence of nonconstant, bounded harmonic functions. The book will be suitable as a textbook for beginning graduate-level courses or independent study by graduate students and advanced undergraduate students in mathematics with a solid grounding in measure theory and a basic familiarity with the elements of group theory. The first seven chapters could also be used as the basis for a short course covering the main results regarding transience/recurrence, decay of return probabilities, and speed. The book has been organized and written so as to be accessible not only to students in probability theory, but also to students whose primary interests are in geometry, ergodic theory, or geometric group theory.
| Author | : Alessandro Figà-Talamanca |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 86 |
| Release | : 1994 |
| Genre | : Mathematics |
| ISBN | : 0821825941 |
This work presents a detailed study of the anisotropic series representations of the free product group Z/2Z*...*Z/2Z. These representations are infinite dimensional, irreducible, and unitary and can be divided into principal and complementary series. Anisotropic series representations are interesting because, while they are not restricted from any larger continuous group in which the discrete group is a lattice, they nonetheless share many properties of such restrictions. The results of this work are also valid for nonabelian free groups on finitely many generators.
| Author | : M. Picardello |
| Publisher | : Cambridge University Press |
| Total Pages | : 378 |
| Release | : 1999-11-18 |
| Genre | : Mathematics |
| ISBN | : 9780521773126 |
Comprehensive and interdisciplinary text covering the interplay between random walks and structure theory.
| Author | : Vadim Kaimanovich |
| Publisher | : Walter de Gruyter |
| Total Pages | : 545 |
| Release | : 2008-08-22 |
| Genre | : Mathematics |
| ISBN | : 3110198088 |
Die jüngsten Entwicklungen zeigen, dass sich Wahrscheinlichkeitsverfahren zu einem sehr wirkungsvollen Werkzeug entwickelt haben, und das auf so unterschiedlichen Gebieten wie statistische Physik, dynamische Systeme, Riemann'sche Geometrie, Gruppentheorie, harmonische Analyse, Graphentheorie und Informatik.
| Author | : Laurent Bartholdi |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 419 |
| Release | : 2006-03-28 |
| Genre | : Mathematics |
| ISBN | : 3764374470 |
This book offers a panorama of recent advances in the theory of infinite groups. It contains survey papers contributed by leading specialists in group theory and other areas of mathematics. Topics include amenable groups, Kaehler groups, automorphism groups of rooted trees, rigidity, C*-algebras, random walks on groups, pro-p groups, Burnside groups, parafree groups, and Fuchsian groups. The accent is put on strong connections between group theory and other areas of mathematics.
| Author | : João Lopes Dias |
| Publisher | : Springer Nature |
| Total Pages | : 184 |
| Release | : 2023-11-29 |
| Genre | : Mathematics |
| ISBN | : 3031413164 |
This volume presents peer-reviewed surveys on new developments in the study of Lyapunov exponents in dynamical systems and its applications to other areas, such as mathematical physics. Written by leading experts in their fields, the contributions are based upon the presentations given by invited speakers at the “New Trends in Lyapunov Exponents” workshop held in Lisbon, Portugal, February 7–11, 2022. The works focus on the concept of Lyapunov exponents in their various manifestations in dynamical systems along with their applications to mathematical physics and other areas of mathematics. The papers reflect the spirit of the conference of promoting new connections among different subjects in dynamical systems. This volume aims primarily at researchers and graduate students working in dynamical systems and related fields, serving as an introduction to active fields of research and as a review of recent results as well.
| Author | : Yves Guivarc'h |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 297 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461224527 |
The concept of symmetric space is of central importance in many branches of mathematics. Compactifications of these spaces have been studied from the points of view of representation theory, geometry, and random walks. This work is devoted to the study of the interrelationships among these various compactifications and, in particular, focuses on the martin compactifications. It is the first exposition to treat compactifications of symmetric spaces systematically and to uniformized the various points of view. The work is largely self-contained, with comprehensive references to the literature. It is an excellent resource for both researchers and graduate students.
| Author | : Semen Grigorʹevich Gindikin |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 372 |
| Release | : 2003 |
| Genre | : Geometry, Differential |
| ISBN | : 9780821834725 |
The book contains survey and research articles devoted mainly to geometry and harmonic analysis of symmetric spaces and to corresponding aspects of group representation theory. The volume is dedicated to the memory of Russian mathematician, F. I. Karpelevich (1927-2000). Of particular interest are the survey articles by Sawyer on the Abel transform on noncompact Riemannian symmetric spaces, and by Anker and Ostellari on estimates for heat kernels on such spaces, as well as thearticle by Bernstein and Gindikin on integral geometry for families of curves. There are also many research papers on topics of current interest. The book is suitable for graduate students and research mathematicians interested in harmonic analysis and representation theory.
| Author | : Michael Cowling |
| Publisher | : Springer |
| Total Pages | : 400 |
| Release | : 2008-02-22 |
| Genre | : Mathematics |
| ISBN | : 3540768920 |
Six leading experts lecture on a wide spectrum of recent results on the subject of the title. They present a survey of various interactions between representation theory and harmonic analysis on semisimple groups and symmetric spaces, and recall the concept of amenability. They further illustrate how representation theory is related to quantum computing; and much more. Taken together, this volume provides both a solid reference and deep insights on current research activity.
