First-passage percolation on the square lattice:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1978
|
| Series: | Lecture notes in mathematics
671 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VIII, 196 S. |
| ISBN: | 3540089284 0387089284 |
Staff View
MARC
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| 245 | 1 | 0 | |a First-passage percolation on the square lattice |c R. T. Smythe ; John C. Wierman |
| 246 | 1 | 3 | |a First passage percolation on the square lattice |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1978 | |
| 300 | |a VIII, 196 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 650 | 4 | |a Limit theorems (Probability theory) | |
| 650 | 4 | |a Matrices | |
| 650 | 4 | |a Renewal theory | |
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Record in the Search Index
| _version_ | 1804116709513101312 |
|---|---|
| adam_text | Table £f Contents
I. Introduction 1
1.1 The percolation model 1
1.2 General formulation of first passage percolation
on L 5
II. Preliminaries 8
2.1 Whitney s theorem and percolation on the sponge • • • 8
2.2 The FKG inequality 12
2.3 Subadditive processes 16
2.4 The Kesten Hammersley Theorem 20
2.5 The connectivity constant 24
2.6 Two results of Hammersley 25
III. Bernoulli percolation 28
3.1 Definition of critical probabilities 29
3.2 p p 31
3 3 pT + p +1 1 32
1 H
3.4 The existence of circuits 35
3.5 Harris1 lower bound 42
3.6 The Seymour Welsh theorem 44
3.7 Hammersley1s upper bound for pH 49
3.8 The number of infinite clusters 50
3.9 The number of clusters per bond and mean cluster
size 51
3.10 Monte Carlo results 62
IV. Definition of the basic processes and the existence of
routes 63
4.1 The basic first passage processes 63
4.2 Integrability results for first passage processes . .67
4.3 Existence of routes 72
4.3.1 Existence of routes for non negative time
coordinates 72
4.3.2 Existence of routes with negative time
coordinates 75
V. Convergence of the first passage processes 78
5.1 Convergence of on and on 78
n n
5.2 Convergence of on 81
n
5.3 Convergence of on and mean convergence 87
n
5.4 Convergence of first passage processes with negative
time coordinates permitted 94
VIII
VI. Renewal theory for percolation processes 101
6.1 Renewal theorems for the first passage processes
when U(0) X 1 101
6.2 Integrability of the renewal process when U(0) p™ .109
VII. The time constant 111
7.1 Conditions for u (U) = 0 111
7.2 Bounds for u(U) 114
7.3 The time constant as a function, of the
distribution U 119
7.3.1 Monotonicity of |i(U) 119
7.3.2 Concavity of u(U@r) 121
7.3.3 Truncation of U 122
VIII. Route length and the height process _¦ . .128
8.1 Convergence a.s. of route lengths when U(0) X . .128
8.2 Numerical estimates for theMroute length ....... .131
3.3 Integrability results for _n when U(0) ~ . . . .135
n
8.4 Route length bounds when U(0) PH 133
3.5 Bounds for the height process when U(0) ~ ... .146
8.6 Height of the cylinder process when U(0) ^ 150
8.7 The height process when U(0) PH 151
8.8 Further results on continuity of n (U) 154
IX. Other percolation models on the square lattice 158
9.1 The first passage process and reach process in the
completely oriented model 159
9.2 The Morgan Welsh model 162
9.3 Asymmetric oriented percolation 167
9.4 A stiff percolation model I67
X. Conjectures and open problems 170
10.1 The determination of u (U) I70
10.2 The rate of convergence to i i(U) 171
10.3 Renewal theory 172
10.4 The constant i i(U) as a functional of U 172
10.5 Route length and the height process 173
10.6 Richardson s model and the value of n(U) 175
Appendix A The FKG inequality 177
Appendix B Ergodic theorems for subadditive processes 183
References 189
Subject Index 194
|
| any_adam_object | 1 |
| author | Smythe, Robert T. Wierman, John C. |
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| dewey-ones | 510 - Mathematics 519 - Probabilities and applied mathematics |
| dewey-raw | 510/.8 519.2 |
| dewey-search | 510/.8 519.2 |
| dewey-sort | 3510 18 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV002253158 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:42:49Z |
| institution | BVB |
| isbn | 3540089284 0387089284 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001480578 |
| oclc_num | 4136062 |
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| physical | VIII, 196 S. |
| publishDate | 1978 |
| publishDateSearch | 1978 |
| publishDateSort | 1978 |
| publisher | Springer |
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| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Smythe, Robert T. Verfasser aut First-passage percolation on the square lattice R. T. Smythe ; John C. Wierman First passage percolation on the square lattice Berlin [u.a.] Springer 1978 VIII, 196 S. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 671 Limiettheorema's gtt Matrices Matrices gtt Percolatietheorie gtt Problèmes limites (Théorie des probabilités) Renouvellement, Théorie du Vernieuwingstheorie gtt Limit theorems (Probability theory) Renewal theory Grenzwertsatz (DE-588)4158163-5 gnd rswk-swf Grenzwertsatz (DE-588)4158163-5 s DE-604 Wierman, John C. Verfasser aut Lecture notes in mathematics 671 (DE-604)BV000676446 671 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001480578&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Smythe, Robert T. Wierman, John C. First-passage percolation on the square lattice Lecture notes in mathematics Limiettheorema's gtt Matrices Matrices gtt Percolatietheorie gtt Problèmes limites (Théorie des probabilités) Renouvellement, Théorie du Vernieuwingstheorie gtt Limit theorems (Probability theory) Renewal theory Grenzwertsatz (DE-588)4158163-5 gnd |
| subject_GND | (DE-588)4158163-5 |
| title | First-passage percolation on the square lattice |
| title_alt | First passage percolation on the square lattice |
| title_auth | First-passage percolation on the square lattice |
| title_exact_search | First-passage percolation on the square lattice |
| title_full | First-passage percolation on the square lattice R. T. Smythe ; John C. Wierman |
| title_fullStr | First-passage percolation on the square lattice R. T. Smythe ; John C. Wierman |
| title_full_unstemmed | First-passage percolation on the square lattice R. T. Smythe ; John C. Wierman |
| title_short | First-passage percolation on the square lattice |
| title_sort | first passage percolation on the square lattice |
| topic | Limiettheorema's gtt Matrices Matrices gtt Percolatietheorie gtt Problèmes limites (Théorie des probabilités) Renouvellement, Théorie du Vernieuwingstheorie gtt Limit theorems (Probability theory) Renewal theory Grenzwertsatz (DE-588)4158163-5 gnd |
| topic_facet | Limiettheorema's Matrices Percolatietheorie Problèmes limites (Théorie des probabilités) Renouvellement, Théorie du Vernieuwingstheorie Limit theorems (Probability theory) Renewal theory Grenzwertsatz |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001480578&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
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