Random walks, boundaries and spectra

Cover......Page 1
Progress in Probability 64......Page 3
Random Walks, Boundaries and Spectra......Page 4
ISBN 9783034602433......Page 5
Contents......Page 8
Preface......Page 10
1. Introduction......Page 28
2. The graph......Page 31
References......Page 41
Introduction......Page 42
1.1. Coxeter groups......Page 45
1.2. Buildings......Page 46
1.3. Random walks and the Hecke algebra......Page 49
2.1. The Hecke algebra as a C∗-algebra......Page 51
2.3. Statement of the Plancherel Theorem for type A2......Page 52
3. The local limit theorem......Page 53
4.1. Root systems and affine Weyl groups......Page 58
4.2. Alcove walks......Page 60
5. Structure of affine Hecke algebras......Page 62
5.1. Bernstein presentation of H......Page 63
5.2. The Macdonald formula......Page 64
5.3. Some representation theory......Page 67
6. Harmonic analysis for the Hecke algebra......Page 68
6.1. The C∗-algebra......Page 69
6.2. A formula for the trace on H 10......Page 70
6.3. Opdam’s trace generating function formula......Page 71
7. The Plancherel Theorem for A2......Page 75
8. Some explicit representations......Page 77
References......Page 78
1. Introduction......Page 82
2.1. Continuity of range for symmetric measures......Page 84
2.2. Discontinuity of range and entropy......Page 85
3. Random walks on Z 
 A......Page 87
4. Symmetric examples of discontinuity of drift......Page 89
References......Page 91
1. Introduction......Page 92
3.1. Definitions......Page 93
3.2. Examples......Page 95
4. Growth and recurrence......Page 96
5. Linear groups over local fields......Page 97
6.1. A framework......Page 98
6.2. An example......Page 99
References......Page 100
1. Introduction......Page 102
2.1. Basic measure theory......Page 105
2.2. Fourier analysis......Page 106
3.1. Mean ergodic theorems......Page 107
3.2. Pointwise ergodic theorems......Page 108
3.3. Maximal inequalities......Page 110
3.4. Rubio de Francia’s maximal inequality......Page 112
4.2. Ergodic theorems for polynomial curves......Page 113
4.4. Regularity of return times......Page 114
References......Page 115
1. Introduction......Page 118
2. Weighted Pascal graphs......Page 119
3. Probability of the first head......Page 123
4. Transformations of weights......Page 125
5. Generalized Stirling triangles......Page 127
5.1. The linear weights......Page 129
5.2. Generalized Stirling-I......Page 130
5.2.2. The case of discrete boundary.......Page 131
5.3. Generalized Stirling-II......Page 133
6. Generalized Eulerian triangles......Page 135
References......Page 136
Resistance Boundaries of Infinite Networks......Page 138
1. Introduction......Page 139
2. The energy space HE......Page 143
2.1. The finitely-supported functions and the harmonic functions......Page 144
2.2. Monopoles......Page 145
3.1. Relating Δ to E......Page 146
3.2. More about monopoles and the spaceM......Page 148
4. Effective resistance......Page 149
4.1.1. Free resistance.......Page 150
4.1.2. Wired resistance.......Page 151
4.2. von Neumann construction of the energy space HE......Page 153
5.1. Gel’fand triples and duality......Page 154
5.2. A Gel’fand triple for HE......Page 156
5.3. The Wiener embedding and the space SE......Page 158
6. Examples......Page 161
Acknowledgement......Page 164
References......Page 165
1. Introduction......Page 170
2.1. Rotationally symmetric manifolds......Page 172
2.2. Curvature conditions for the Liouville property......Page 175
3.1. Boundaries at infinity......Page 176
3.2. Angular convergence and solvability of the Dirichlet problem at infinity......Page 177
3.3. The role of lower curvature bounds......Page 179
4.1. Description of the manifold......Page 181
4.2. Asymptotic behaviour of Brownian motion......Page 183
4.3. Description of the Poisson boundary......Page 184
References......Page 186
1. Introduction......Page 190
2. Stochastic incompleteness......Page 191
3. Stochastically incomplete manifolds......Page 193
4. Stochastically incomplete graphs......Page 194
4.1. Stochastically incomplete subgraphs......Page 197
4.2. Spherically symmetric graphs......Page 199
5.1. Volume growth......Page 203
Acknowledgment......Page 204
References......Page 205
1. Introduction......Page 208
2. Definitions and preliminaries......Page 210
3. The Allegretto-Piepenbrink theorem......Page 212
3.1. The ground state representation......Page 213
3.2. A Harnack inequality......Page 214
3.3. Two limiting procedures......Page 216
3.4. Proof of the Allegretto-Piepenbrink theorem......Page 217
4.1. Two notions of boundary and Shnol’s theorem......Page 218
4.2. Proof of the Shnol’ type theorems......Page 220
4.3. Bounded Laplacians in the magnifying glass......Page 223
5. Non-regular Dirichlet forms – a short discussion......Page 224
References......Page 225
1. Introduction......Page 228
2. Setup......Page 229
3. One-dimensional graph, N0......Page 233
4. General graphs......Page 238
5. Trees......Page 240
6. Strongly correlated random potential on a tree......Page 246
7. Loop tree models......Page 247
7.1. Regular loop tree model......Page 248
7.2. Mean-field loop model......Page 250
Acknowledgment......Page 251
References......Page 252
2. Countable groups......Page 254
3. Locally finite groups and ultra-metrics......Page 255
4. The spectral distribution and the isospectral profileof a Laplace operator on a countable group......Page 257
5. The relationship between the spectral distribution and the isospectral profile......Page 258
5.1. Finitely generated groups and measures with finite second moment......Page 259
5.2. Countable locally finite groups and series of Haar measures......Page 260
References......Page 261
1.1. Graphs......Page 262
1.2. The adjacency operator and Laplacians......Page 263
1.3. Amenable groups and their Cayley graphs......Page 265
2.1. Percolation......Page 267
2.2. The integrated density of states......Page 270
2.3. The integer lattice......Page 272
2.4. The regular infinite tree (Bethe lattice)......Page 276
2.5. Equality and non-equality of Lifshits and van Hove exponents on amenable Cayley graphs......Page 279
2.6. Outlook: some further models......Page 281
References......Page 282
1.1. Phase transitions in Inhomogeneous random graphs......Page 286
1.2. Inhomogeneous random graphs......Page 287
2.1. The breadth-first walk for inhomogeneous random graph and branching processes......Page 289
2.2.1. Survival probability.......Page 290
2.2.2. Total progeny.......Page 291
3. Supercritical phase: Tκ > 1......Page 293
4.1. Rank 1 model: sufficient conditions for the log n scaling......Page 294
5.1. Model Gn.c/n......Page 296
5.2. Rank 1 case on i.i.d. with finite 3d moment......Page 297
5.3. Rank 1 case with infinite 3d moment......Page 298
6. Spectra of rank 1 graphs......Page 300
References......Page 301
1.1. The Ising model......Page 304
1.2. Groups of automorphisms of rooted regular trees......Page 307
1.3. Plan of the paper......Page 308
2.1. Grigorchuk’s group......Page 309
2.2. The Basilica group......Page 310
3.1. Hanoi Towers group H(3)......Page 314
3.2. The Sierpiński gasket......Page 318
3.3. Renormalization approach......Page 320
4.1. The Schreier graphs of the Grigorchuk’s group......Page 321
4.2. The Schreier graphs of the Basilica group......Page 323
4.3. The Schreier graphs of H(3)......Page 324
4.4. The Sierpi´nski graphs......Page 325
4.5. Correspondences via Fisher’s Theorem......Page 328
Acknowledgment......Page 329
References......Page 330
1. Introduction......Page 332
2. Asymptotics for a fixed symbol......Page 334
3. Transition asymptotics......Page 339
4. Asymptotics for Fredholm determinants......Page 341
References......Page 345