Random Walks on Reductive Groups

Authors: Yves Benoist, Jean-François Quint



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

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.

Table of contents (17 chapters)

Front Matter

Introduction

The Law of Large Numbers
• Front Matter
• Stationary Measures
• The Law of Large Numbers
• Linear Random Walks
• Finite Index Subsemigroups

Reductive Groups
• Front Matter
• Loxodromic Elements
• The Jordan Projection of Semigroups
• Reductive Groups and Their Representations
• Zariski Dense Subsemigroups
• Random Walks on Reductive Groups

The Central Limit Theorem
• Front Matter
• Transfer Operators over Contracting Actions
• Limit Laws for Cocycles
• Limit Laws for Products of Random Matrices
• Regularity of the Stationary Measure

The Local Limit Theorem
• Front Matter
• The Spectrum of the Complex Transfer Operator
• The Local Limit Theorem for Cocycles
• The Local Limit Theorem for Products of Random Matrices

Back Matter

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