Random Walks on Reductive Groups

Description

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. Front Matter....Pages I-XI Introduction....Pages 1-16 Front Matter....Pages 17-17 Stationary Measures....Pages 19-36 The Law of Large Numbers....Pages 37-49 Linear Random Walks....Pages 51-75 Finite Index Subsemigroups....Pages 77-86 Front Matter....Pages 87-87 Loxodromic Elements....Pages 89-113 The Jordan Projection of Semigroups....Pages 115-126 Reductive Groups and Their Representations....Pages 127-145 Zariski Dense Subsemigroups....Pages 147-152 Random Walks on Reductive Groups....Pages 153-167 Front Matter....Pages 169-169 Transfer Operators over Contracting Actions....Pages 171-189 Limit Laws for Cocycles....Pages 191-202 Limit Laws for Products of Random Matrices....Pages 203-222 Regularity of the Stationary Measure....Pages 223-245 Front Matter....Pages 247-247 The Spectrum of the Complex Transfer Operator....Pages 249-258 The Local Limit Theorem for Cocycles....Pages 259-272 The Local Limit Theorem for Products of Random Matrices....Pages 273-285 Back Matter....Pages 287-323