Percolation:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin [u.a.]
Springer
2010
|
| Edition: | Second edition, [Nachdr.] |
| Series: | Grundlehren der mathematischen Wissenschaften
321 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Text in englischer Sprache Softcover version of original hardcover ed. 1999 |
| Physical Description: | XIII, 444 S. Ill., graph. Darst. |
| ISBN: | 9783540649021 9783642084423 |
Staff View
MARC
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| 100 | 1 | |a Grimmett, Geoffrey |d 1950- |e Verfasser |0 (DE-588)120872919 |4 aut | |
| 245 | 1 | 0 | |a Percolation |c Geoffrey Grimmett |
| 250 | |a Second edition, [Nachdr.] | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2010 | |
| 300 | |a XIII, 444 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften |v 321 | |
| 490 | 0 | |a A Series of Comprehensive Studies in Mathematics | |
| 500 | |a Text in englischer Sprache | ||
| 500 | |a Softcover version of original hardcover ed. 1999 | ||
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Record in the Search Index
| _version_ | 1804143806042341376 |
|---|---|
| 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
к
........................................ 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
Subcriticai
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
Analyticky
offe
and
χ......................................
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 aFinite 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
Θ,
xf, and
к
..............................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 Aß
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 | BV037212404 |
| classification_rvk | SK 370 SK 820 UG 3100 UG 3700 |
| classification_tum | MAT 606f MAT 600f PHY 064f |
| ctrlnum | (OCoLC)711808063 (DE-599)BVBBV037212404 |
| discipline | Physik Mathematik |
| edition | Second edition, [Nachdr.] |
| format | Book |
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| id | DE-604.BV037212404 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:53:30Z |
| institution | BVB |
| isbn | 9783540649021 9783642084423 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-021126431 |
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| physical | XIII, 444 S. Ill., graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics |
| spelling | Grimmett, Geoffrey 1950- Verfasser (DE-588)120872919 aut Percolation Geoffrey Grimmett Second edition, [Nachdr.] Berlin [u.a.] Springer 2010 XIII, 444 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 321 A Series of Comprehensive Studies in Mathematics Text in englischer Sprache Softcover version of original hardcover ed. 1999 Perkolation (DE-588)4115530-0 gnd rswk-swf Statistische Physik (DE-588)4057000-9 gnd rswk-swf Perkolationstheorie (DE-588)4323583-9 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 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021126431&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 Statistische Physik (DE-588)4057000-9 gnd Perkolationstheorie (DE-588)4323583-9 gnd |
| subject_GND | (DE-588)4115530-0 (DE-588)4057000-9 (DE-588)4323583-9 |
| 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 Statistische Physik (DE-588)4057000-9 gnd Perkolationstheorie (DE-588)4323583-9 gnd |
| topic_facet | Perkolation Statistische Physik Perkolationstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021126431&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT grimmettgeoffrey percolation |