Random walks on infinite graphs and groups:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2008
|
| Edition: | 1. publ., digital print. with corr. |
| Series: | Cambridge tracts in mathematics
138 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 334 S. graph. Darst. |
| ISBN: | 9780521061728 9780521552929 |
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MARC
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| 245 | 1 | 0 | |a Random walks on infinite graphs and groups |c Wolfgang Woess |
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Record in the Search Index
| _version_ | 1804143769002442752 |
|---|---|
| adam_text | CONTENTS
Preface viii
Errata xii
Chapter I. The type problem
1
1.
Basic facts
1
A. Pólya s
walk
1
B. Irreducible Markov chains
2
C. Random walks on graphs
7
D. Trees
9
E. Random walks on finitely generated groups
10
F. Locally finite graphs and
topologica!
groups
12
2.
Recurrence and transience of infinite networks
14
A. Reversible Markov chains
14
B. Flows, capacity, and Nash-Williams criterion
18
C. Comparison with non-reversible Markov chains
23
3.
Applications to random walks
25
A. Comparison theorems for random walks on graphs
25
B. Growth and the classification of recurrent groups
30
C. Random walks on quasi-transitive graphs
36
4.
Isoperimetric inequalities
39
A. Isoperimetric and Sobolev inequalities
39
B. Cartesian products
43
C. Isoperimetric inequalities and growth
45
5.
Transient subtrees, and the classification of the recurrent
quasi-transitive graphs
49
A. Transient subtrees
49
B. Transient subtrees in quasi-transitive graphs
54
6.
More on recurrence
56
A. Generalized lattices
56
B. More on trees
62
С
Extremal length and plane tilings
67
D. Circle packings and random walks
71
Notes and remarks
77
Chapter II. The spectral radius
81
7.
Superharmonic functions and p-recurrence
81
A. The spectra! radius and superharmonic functions
81
B. /^Recurrence
82
8.
The spectral radius, the rate of escape, and generalized
lattices
84
Contents
A. The rate of escape
84
B. Application to generalized lattices
88
9.
Computing the Green function
93
A. Singularities of the Green function
93
B. A functional equation
98
C. Free products
101
10.
The spectral radius and strong isoperimetric
inequalities
110
A. The spectral radius of reversible Markov chains
110
B. Application to random walks on graphs
112
C. Examples: trees, strongly ramified graphs, and
tilings
114
11.
A lower bound for simple random walks
118
12.
The spectral radius and amenability
123
A. Amenable groups
123
B. Automorphism groups and the spectral radius
125
C. Some explicit computations
129
Notes and remarks
136
Chapter III. The asymptotic behaviour of transition
probabilities
139
13.
The local central limit theorem on the grid
139
14.
Growth, isoperimetric inequalities, and the asymptotic
type of random walk
145
A. Upper bounds and Nash inequalities
146
B. Gaussian upper bounds
152
C. Lower bounds
157
15.
The asymptotic type of random walks on amenable
groups
160
A. Comparison and stability of asymptotic type on
groups
160
B. Polycyclic groups
164
С
The solvable Baumslag-Solitar groups
168
D. Random walks on lamplighter groups
169
16.
Simple random walks on the Sierpinski graphs
171
A. Stopping times and an equation for the Green
function
172
B. Singularity analysis
176
17.
Local limit theorems on free products
181
A. The typical case:
n~3¡2
183
B.
Instability of the exponent
189
18.
Intermezzo: Cartesian products
195
Contents
19.
Free groups and homogeneous trees
199
A. Space-time asymptotics for aperiodic simple random
walks on Tm
199
B. Finite range random walks on free groups
205
C. Radial random walks on the homogeneous tree
213
Notes and remarks
216
Chapter
IV.
An introduction to topological
boundary theory
220
20.
A probabilistic approach to the Dirichlet problem, and
a class of compactifications
220
A. The Dirichlet problem and convergence to the
boundary
220
B. Compactifications with hyperbolic properties
224
21.
Ends of graphs and the Dirichlet problem
230
A. The transitive case
232
B. Geometric adaptedness conditions
239
22.
Hyperbolic graphs and groups
242
23.
The Dirichlet problem for circle packing graphs
252
24.
The construction of the Martin boundary
256
25.
Generalized lattices, Abelian and
nilpotent
groups, and
graphs with polynomial growth
262
A. Exponentials and extended exponentials
262
B. The Martin compactification of random walks on
the grid
268
26.
Trees, ends, and free products
275
A. Thin ends and trees
277
B. Free products
283
27.
The Martin boundary of hyperbolic graphs
288
28.
Cartesian products
297
A. Minimal harmonic functions on Cartesian
products
297
B. The Martin compactification of
Τ χ Ζ
301
Notes and remarks
309
Acknowledgments
315
Bibliography
316
Index
331
|
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| author | Woess, Wolfgang 1954- |
| author_GND | (DE-588)124183077 |
| author_facet | Woess, Wolfgang 1954- |
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| author_sort | Woess, Wolfgang 1954- |
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| bvnumber | BV037185476 |
| classification_rvk | SK 820 |
| classification_tum | MAT 055f MAT 605f |
| ctrlnum | (OCoLC)746305540 (DE-599)BVBBV037185476 |
| discipline | Mathematik |
| edition | 1. publ., digital print. with corr. |
| format | Book |
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| id | DE-604.BV037185476 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:52:55Z |
| institution | BVB |
| isbn | 9780521061728 9780521552929 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-021100003 |
| oclc_num | 746305540 |
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| owner_facet | DE-355 DE-BY-UBR DE-188 DE-824 |
| physical | XIV, 334 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Cambridge tracts in mathematics |
| series2 | Cambridge tracts in mathematics |
| spelling | Woess, Wolfgang 1954- Verfasser (DE-588)124183077 aut Random walks on infinite graphs and groups Wolfgang Woess 1. publ., digital print. with corr. Cambridge [u.a.] Cambridge Univ. Press 2008 XIV, 334 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge tracts in mathematics 138 Unendlicher Graph (DE-588)4390888-3 gnd rswk-swf Unendliche Gruppe (DE-588)4375539-2 gnd rswk-swf Irrfahrt (DE-588)4561504-4 gnd rswk-swf Unendlicher Graph (DE-588)4390888-3 s Unendliche Gruppe (DE-588)4375539-2 s Irrfahrt (DE-588)4561504-4 s DE-604 Cambridge tracts in mathematics 138 (DE-604)BV000000001 138 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021100003&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Woess, Wolfgang 1954- Random walks on infinite graphs and groups Cambridge tracts in mathematics Unendlicher Graph (DE-588)4390888-3 gnd Unendliche Gruppe (DE-588)4375539-2 gnd Irrfahrt (DE-588)4561504-4 gnd |
| subject_GND | (DE-588)4390888-3 (DE-588)4375539-2 (DE-588)4561504-4 |
| title | Random walks on infinite graphs and groups |
| title_auth | Random walks on infinite graphs and groups |
| title_exact_search | Random walks on infinite graphs and groups |
| title_full | Random walks on infinite graphs and groups Wolfgang Woess |
| title_fullStr | Random walks on infinite graphs and groups Wolfgang Woess |
| title_full_unstemmed | Random walks on infinite graphs and groups Wolfgang Woess |
| title_short | Random walks on infinite graphs and groups |
| title_sort | random walks on infinite graphs and groups |
| topic | Unendlicher Graph (DE-588)4390888-3 gnd Unendliche Gruppe (DE-588)4375539-2 gnd Irrfahrt (DE-588)4561504-4 gnd |
| topic_facet | Unendlicher Graph Unendliche Gruppe Irrfahrt |
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