Probability on graphs: random processes on graphs and lattices
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge, United Kingdom ; New York, NY ; Port Melbourne, Australia ; New Delhi, India ; Singapore
Cambridge University Press
2018
|
| Edition: | Second edition |
| Series: | Textbooks / Institute of Mathematical Statistics
8 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Item Description: | Enthält Literaturverzeichnis (Seite [240] - 260) und Index |
| Physical Description: | xi, 265 Seiten Diagramme, Illustrationen |
| ISBN: | 9781108438179 |
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| 245 | 1 | 0 | |a Probability on graphs |b random processes on graphs and lattices |c Geoffrey Grimmett (Statistical Laboratory, University of Cambridge) |
| 250 | |a Second edition | ||
| 264 | 1 | |a Cambridge, United Kingdom ; New York, NY ; Port Melbourne, Australia ; New Delhi, India ; Singapore |b Cambridge University Press |c 2018 | |
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Record in the Search Index
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|---|---|
| adam_text | ix
1
1
3
8
11
14
16
18
21
21
23
28
31
32
36
39
39
42
45
53
53
56
59
59
62
63
Contents
Preface
Random Walks on Graphs
1.1 Random Walks and Reversible Markov Chains
1.2 Electrical Networks
1.3 Rows and Energy
1.4 Recurrence and Resistance
1.5 Polya’s Theorem
1.6 Graph Theory
1.7 Exercises
Uniform Spanning Tree
2.1 Definition
2.2 Wilson’s Algorithm
2.3 Weak Limits on Lattices
2.4 Uniform Forest
2.5 Schramm—Lowner Evolutions
2.6 Exercises
Percolation and Self-Avoiding Walks
3.1 Percolation and Phase Transition
3.2 Self-Avoiding Walks
3.3 Connective Constant of the Hexagonal Lattice
3.4 Coupled Percolation
3.5 Oriented Percolation
3.6 Exercises
Association and Influence
4.1 Holley Inequality
4.2 FKG Inequality
4.3 BK Inequality
65
67
72
80
83
86
86
90
96
99
103
107
115
126
130
131
131
133
135
137
139
142
145
146
146
148
152
154
156
156
159
160
165
167
173
183
186
Contents
4.4 Hoeffding Inequality
4.5 Influence for Product Measures
4.6 Proofs of Influence Theorems
4.7 Russo’s Formula and Sharp Thresholds
4.8 Exercises
Further Percolation
5.1 Subcritical Phase
5.2 Supercritical Phase
5.3 Uniqueness of the Infinite Cluster
5.4 Phase Transition
5.5 Open Paths in Annuli
5.6 The Critical Probability in Two Dimensions
5.7 Cardy’s Formula
5.8 The Critical Probability via the Sharp-Threshold Theorem
5.9 Exercises
Contact Process
6.1 Stochastic Epidemics
6.2 Coupling and Duality
6.3 Invariant Measures and Percolation
6.4 The Critical Value
6.5 The Contact Model on a Tree
6.6 Space-Time Percolation
6.7 Exercises
Gibbs States
7.1 Dependency Graphs
7.2 Markov and Gibbs Random Fields
7.3 Ising and Potts Models
7.4 Exercises
Random-Cluster Model
8.1 The Random-Cluster, Ising, and Potts Models
8.2 Basic Properties
8.3 Infinite-Volume Limits and Phase Transition
8.4 Open Problems
8.5 In Two Dimensions
8.6 Proof of the Critical Point in Two Dimensions
8.7 Random Even Graphs
8.8 Exercises
Contents vii
9 Quantum Ising Model 189
9.1 The Model 189
9.2 Continuum Random-Cluster Model 191
9.3 Quantum Ising via Random-Cluster Model 193
9.4 Long-Range Order 198
9.5 Entanglement in One Dimension 199
9.6 Exercises 203
10 Interacting Particle Systems 204
10.1 Introductory Remarks 204
10.2 Contact Model 206
10.3 Voter Model 207
10.4 Exclusion Model 211
10.5 Stochastic Ising Model 214
10.6 Exercises 218
11 Random Graphs 219
11.1 Erdos—Renyi Graphs 219
11.2 Giant Component 221
11.3 Independence and Colouring 226
11.4 Exercises 231
12 Lorentz Gas 233
12.1 Lorentz Model 233
12.2 The Square Lorentz Gas 234
12.3 In the Plane 237
12.4 Exercises 238
References 240
Index 261
ix
1
1
3
8
11
14
16
18
21
21
23
28
31
32
36
39
39
42
45
53
53
56
59
59
62
63
Contents
Preface
Random Walks on Graphs
1.1 Random Walks and Reversible Markov Chains
1.2 Electrical Networks
1.3 Rows and Energy
1.4 Recurrence and Resistance
1.5 Polya’s Theorem
1.6 Graph Theory
1.7 Exercises
Uniform Spanning Tree
2.1 Definition
2.2 Wilson’s Algorithm
2.3 Weak Limits on Lattices
2.4 Uniform Forest
2.5 Schramm—Lowner Evolutions
2.6 Exercises
Percolation and Self-Avoiding Walks
3.1 Percolation and Phase Transition
3.2 Self-Avoiding Walks
3.3 Connective Constant of the Hexagonal Lattice
3.4 Coupled Percolation
3.5 Oriented Percolation
3.6 Exercises
Association and Influence
4.1 Holley Inequality
4.2 FKG Inequality
4.3 BK Inequality
65
67
72
80
83
86
86
90
96
99
103
107
115
126
130
131
131
133
135
137
139
142
145
146
146
148
152
154
156
156
159
160
165
167
173
183
186
Contents
4.4 Hoeffding Inequality
4.5 Influence for Product Measures
4.6 Proofs of Influence Theorems
4.7 Russo’s Formula and Sharp Thresholds
4.8 Exercises
Further Percolation
5.1 Subcritical Phase
5.2 Supercritical Phase
5.3 Uniqueness of the Infinite Cluster
5.4 Phase Transition
5.5 Open Paths in Annuli
5.6 The Critical Probability in Two Dimensions
5.7 Cardy’s Formula
5.8 The Critical Probability via the Sharp-Threshold Theorem
5.9 Exercises
Contact Process
6.1 Stochastic Epidemics
6.2 Coupling and Duality
6.3 Invariant Measures and Percolation
6.4 The Critical Value
6.5 The Contact Model on a Tree
6.6 Space-Time Percolation
6.7 Exercises
Gibbs States
7.1 Dependency Graphs
7.2 Markov and Gibbs Random Fields
7.3 Ising and Potts Models
7.4 Exercises
Random-Cluster Model
8.1 The Random-Cluster, Ising, and Potts Models
8.2 Basic Properties
8.3 Infinite-Volume Limits and Phase Transition
8.4 Open Problems
8.5 In Two Dimensions
8.6 Proof of the Critical Point in Two Dimensions
8.7 Random Even Graphs
8.8 Exercises
Contents vii
9 Quantum Ising Model 189
9.1 The Model 189
9.2 Continuum Random-Cluster Model 191
9.3 Quantum Ising via Random-Cluster Model 193
9.4 Long-Range Order 198
9.5 Entanglement in One Dimension 199
9.6 Exercises 203
10 Interacting Particle Systems 204
10.1 Introductory Remarks 204
10.2 Contact Model 206
10.3 Voter Model 207
10.4 Exclusion Model 211
10.5 Stochastic Ising Model 214
10.6 Exercises 218
11 Random Graphs 219
11.1 Erdos—Renyi Graphs 219
11.2 Giant Component 221
11.3 Independence and Colouring 226
11.4 Exercises 231
12 Lorentz Gas 233
12.1 Lorentz Model 233
12.2 The Square Lorentz Gas 234
12.3 In the Plane 237
12.4 Exercises 238
References 240
Index 261
|
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| ctrlnum | (OCoLC)1024100488 (DE-599)BVBBV044482906 |
| discipline | Mathematik |
| edition | Second edition |
| format | Book |
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| indexdate | 2024-07-10T07:54:11Z |
| institution | BVB |
| isbn | 9781108438179 |
| language | English |
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| spelling | Grimmett, Geoffrey 1950- Verfasser (DE-588)120872919 aut Probability on graphs random processes on graphs and lattices Geoffrey Grimmett (Statistical Laboratory, University of Cambridge) Second edition Cambridge, United Kingdom ; New York, NY ; Port Melbourne, Australia ; New Delhi, India ; Singapore Cambridge University Press 2018 xi, 265 Seiten Diagramme, Illustrationen txt rdacontent n rdamedia nc rdacarrier Textbooks / Institute of Mathematical Statistics 8 Enthält Literaturverzeichnis (Seite [240] - 260) und Index Zufallsgraph (DE-588)4277661-2 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 s Zufallsgraph (DE-588)4277661-2 s DE-604 Institute of Mathematical Statistics Textbooks 8 (DE-604)BV036598560 8 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029883065&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029883065&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Grimmett, Geoffrey 1950- Probability on graphs random processes on graphs and lattices Zufallsgraph (DE-588)4277661-2 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| subject_GND | (DE-588)4277661-2 (DE-588)4057630-9 |
| title | Probability on graphs random processes on graphs and lattices |
| title_auth | Probability on graphs random processes on graphs and lattices |
| title_exact_search | Probability on graphs random processes on graphs and lattices |
| title_full | Probability on graphs random processes on graphs and lattices Geoffrey Grimmett (Statistical Laboratory, University of Cambridge) |
| title_fullStr | Probability on graphs random processes on graphs and lattices Geoffrey Grimmett (Statistical Laboratory, University of Cambridge) |
| title_full_unstemmed | Probability on graphs random processes on graphs and lattices Geoffrey Grimmett (Statistical Laboratory, University of Cambridge) |
| title_short | Probability on graphs |
| title_sort | probability on graphs random processes on graphs and lattices |
| title_sub | random processes on graphs and lattices |
| topic | Zufallsgraph (DE-588)4277661-2 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| topic_facet | Zufallsgraph Stochastischer Prozess |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029883065&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029883065&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV036598560 |
| work_keys_str_mv | AT grimmettgeoffrey probabilityongraphsrandomprocessesongraphsandlattices |
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