Percolation
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| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
Berlin ; Heidelberg
Springer
[1999]
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| Ausgabe: | second edition |
| Schriftenreihe: | Grundlehren der mathematischen Wissenschaften
321 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
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MARC
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| 100 | 1 | |a Grimmett, Geoffrey |d 1950- |0 (DE-588)120872919 |4 aut | |
| 245 | 1 | 0 | |a Percolation |c Geoffrey Grimmett |
| 250 | |a second edition | ||
| 264 | 1 | |a Berlin ; Heidelberg |b Springer |c [1999] | |
| 264 | 4 | |c © 1999 | |
| 300 | |a xiii, 444 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften |v 321 | |
| 650 | 0 | 7 | |a Perkolation |0 (DE-588)4115530-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Perkolationstheorie |0 (DE-588)4323583-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Statistische Physik |0 (DE-588)4057000-9 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
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| adam_text |
Contents
1 What is Percolation? 1
1.1 Modelling a Random Medium 1
1.2 Why Percolation? 3
1.3 Bond Percolation 9
1.4 The Critical Phenomenon 13
1.5 The Main Questions 20
1.6 Site Percolation 24
1.7 Notes 29
2 Some Basic Techniques 32
2.1 Increasing Events 32
2.2 The FKG Inequality 34
2.3 The BK Inequality 37
2.4 Russo's Formula 41
2.5 Inequalities of Reliability Theory 46
2.6 Another Inequality 49
2.7 Notes 51
3 Critical Probabilities 53
3.1 Equalities and Inequalities 53
3.2 Strict Inequalities 57
3.3 Enhancements 63
3.4 Bond and Site Critical Probabilities 71
3.5 Notes 75
4 The Number of Open Clusters per Vertex 77
4.1 Definition 77
4.2 Lattice Animals and Large Deviations 79
4.3 Differentiability of k 84
4.4 Notes 86
xii Contents
5 Exponential Decay 87
5.1 Mean Cluster Size 87
5.2 Exponential Decay of the Radius Distribution beneath pc 88
5.3 Using Differential Inequalities 102
5.4 Notes 114
6 The Subcritical Phase 117
6.1 The Radius of an Open Cluster 117
6.2 Connectivity Functions and Correlation Length 126
6.3 Exponential Decay of the Cluster Size Distribution 132
6.4 Analyticity of k and x 142
6.5 Notes 144
7 Dynamic and Static Renormalization 146
7.1 Percolation in Slabs 146
7.2 Percolation of Blocks 148
7.3 Percolation in Half Spaces 162
7.4 Static Renormalization 176
7.5 Notes 196
8 The Supercritical Phase 197
8.1 Introduction 197
8.2 Uniqueness of the Infinite Open Cluster 198
8.3 Continuity of the Percolation Probability 202
8.4 The Radius of a Finite Open Cluster 205
8.5 Truncated Connectivity Functions and Correlation Length 213
8.6 Sub Exponential Decay of the Cluster Size Distribution 215
8.7 Differentiability of 6, /f, and k 224
8.8 Geometry of the Infinite Open Cluster 226
8.9 Notes 229
9 Near the Critical Point: Scaling Theory 232
9.1 Power Laws and Critical Exponents 232
9.2 Scaling Theory 239
9.3 Renormalization 244
9.4 The Incipient Infinite Cluster 249
9.5 Notes 252
10 Near the Critical Point: Rigorous Results 254
10.1 Percolation on a Tree 254
10.2 Inequalities for Critical Exponents 262
10.3 Mean Field Theory 269
10.4 Notes 278
Contents xiii
11 Bond Percolation in Two Dimensions 281
11.1 Introduction 281
11.2 Planar Duality 283
11.3 The Critical Probability Equals \ 287
11.4 Tail Estimates in the Supercritical Phase 295
11.5 Percolation on Subsets of the Square Lattice 303
11.6 Central Limit Theorems 309
11.7 Open Circuits in Annuli 314
11.8 Power Law Inequalities 324
11.9 Inhomogeneous Square and Triangular Lattices 331
11.10 Notes 345
12 Extensions of Percolation 349
12.1 Mixed Percolation on a General Lattice 349
12.2 AB Percolation 351
12.3 Long Range Percolation in One Dimension 351
12.4 Surfaces in Three Dimensions 359
12.5 Entanglement in Percolation 362
12.6 Rigidity in Percolation 364
12.7 Invasion Percolation 366
12.8 Oriented Percolation 367
12.9 First Passage Percolation 369
12.10 Continuum Percolation 371
13 Percolative Systems 378
13.1 Capacitated Networks 378
13.2 Random Electrical Networks 380
13.3 Stochastic Pin Ball 382
13.4 Fractal Percolation 385
13.5 Contact Model 390
13.6 Random Cluster Model 393
Appendix I. The Infinite Volume Limit for Percolation 397
Appendix II. The Subadditive Inequality 399
List of Notation 401
References 404
Index of Names 435
Subject Index 439 |
| any_adam_object | 1 |
| author | Grimmett, Geoffrey 1950- |
| author_GND | (DE-588)120872919 |
| author_facet | Grimmett, Geoffrey 1950- |
| author_role | aut |
| author_sort | Grimmett, Geoffrey 1950- |
| author_variant | g g gg |
| building | Verbundindex |
| bvnumber | BV012507777 |
| classification_rvk | SK 370 SK 820 UG 3100 |
| classification_tum | MAT 600f PHY 064f MAT 606f |
| ctrlnum | (OCoLC)441356156 (DE-599)BVBBV012507777 |
| dewey-full | 530.13 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.13 |
| dewey-search | 530.13 |
| dewey-sort | 3530.13 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| edition | second edition |
| format | Book |
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| id | DE-604.BV012507777 |
| illustrated | Illustrated |
| indexdate | 2025-01-27T14:19:50Z |
| institution | BVB |
| isbn | 3540649026 9783642084423 9783540649021 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008489689 |
| oclc_num | 441356156 |
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| physical | xiii, 444 Seiten Illustrationen |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Grimmett, Geoffrey 1950- (DE-588)120872919 aut Percolation Geoffrey Grimmett second edition Berlin ; Heidelberg Springer [1999] © 1999 xiii, 444 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 321 Perkolation (DE-588)4115530-0 gnd rswk-swf Perkolationstheorie (DE-588)4323583-9 gnd rswk-swf Statistische Physik (DE-588)4057000-9 gnd rswk-swf Perkolationstheorie (DE-588)4323583-9 s DE-604 Perkolation (DE-588)4115530-0 s Statistische Physik (DE-588)4057000-9 s Grundlehren der mathematischen Wissenschaften 321 (DE-604)BV000000395 321 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008489689&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Grimmett, Geoffrey 1950- Percolation Grundlehren der mathematischen Wissenschaften Perkolation (DE-588)4115530-0 gnd Perkolationstheorie (DE-588)4323583-9 gnd Statistische Physik (DE-588)4057000-9 gnd |
| subject_GND | (DE-588)4115530-0 (DE-588)4323583-9 (DE-588)4057000-9 |
| title | Percolation |
| title_auth | Percolation |
| title_exact_search | Percolation |
| title_full | Percolation Geoffrey Grimmett |
| title_fullStr | Percolation Geoffrey Grimmett |
| title_full_unstemmed | Percolation Geoffrey Grimmett |
| title_short | Percolation |
| title_sort | percolation |
| topic | Perkolation (DE-588)4115530-0 gnd Perkolationstheorie (DE-588)4323583-9 gnd Statistische Physik (DE-588)4057000-9 gnd |
| topic_facet | Perkolation Perkolationstheorie Statistische Physik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008489689&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT grimmettgeoffrey percolation |