The Wulff crystal in ising and percolation models: Ecole d'Eté de Probabilités de Saint-Flour XXXIV - 2004
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2006
|
| Series: | Lecture notes in mathematics
1878 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 264 S. Ill., graph. Darst. 235 mm x 155 mm |
| ISBN: | 3540309888 9783540309888 |
Staff View
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| 245 | 1 | 0 | |a The Wulff crystal in ising and percolation models |b Ecole d'Eté de Probabilités de Saint-Flour XXXIV - 2004 |c Raphaël Cerf |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2006 | |
| 300 | |a XIV, 264 S. |b Ill., graph. Darst. |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 1878 | |
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| 650 | 7 | |a Percolatietheorie |2 gtt | |
| 650 | 4 | |a Transformations de phase (Physique statistique) | |
| 650 | 4 | |a Ising model | |
| 650 | 4 | |a Percolation (Statistical physics) | |
| 650 | 4 | |a Phase transformations (Statistical physics) | |
| 650 | 4 | |a Wulff construction (Statistical physics) | |
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Record in the Search Index
| _version_ | 1805066766644150272 |
|---|---|
| adam_text |
Contents
Part I Introduction
1
Phase coexistence and subadditivity
. 3
1.1
Water and oil
. 3
1.2
Subadditivity
. 5
1.3
Cramer's theorem
. 6
Part II Presentation of the models
2
Ising model
. 15
2.1
Construction of the model
. 15
2.2
First asymptotics
. 16
2.3
Phase transition
. 18
2.4
Proofs of the heuristics
. 18
3
Bernoulli percolation
. 25
3.1
The probability space
. 25
3.2
Order
onß. 26
3.3
Phase transition
. 26
4
FK or random cluster model
. 31
4.1
Finite volume FK measures
. 31
4.2
Phase transition
. 32
4.3
FK Ising coupling
. 33
4.4
Boundary conditions
. 41
XII Contents
Part III Main results
The
Wulff
crystal
. 45
5.1
Ising model
. 45
5.2
Bernoulli percolation
. 55
5.3
FK percolation
. 58
5.4
What do we know about the
Wulff
crystal?
. 60
5.5
Bibliographical comments
. 62
Part IV Large deviation principles
6
Large deviation theory
. 67
6.1
Main definitions
. 67
6.2
/-tightness
. 68
6.3
Contraction principle
. 72
6.4
Varadhan's lemma
. 73
7
Surface large deviation principles
. 75
7.1
Surface energy
. 75
7.2
The empirical magnetization
. 78
7.3
Minimal surfaces
. 79
7.4
The cluster shapes
. 82
7.5
FK percolation
. 83
8
Volume large deviations
. 85
8.1
Bernoulli percolation
. 85
8.2
FK percolation
. 92
8.3
Ising model
. 96
Part V Fundamental probabilistic estimates
9
Coarse graining
.105
9.1
The good blocks
.105
9.2
Extension to FK measures
.110
9.3
The rescaled lattice
.112
9.4
Two rough estimates
.114
10
Decoupling
.117
10.1
Half-space clusters
.117
10.2
Decoupling lemma
.121
Contents XIII
11
Surface tension
.129
11.1
Existence
.129
11.2
Finite
volume
definition
.133
11.3
Basic
properties
.136
11.4
Separating sets
.141
11.5
What do we know about the surface tension?
.145
12
Interface estimate
.147
12.1
Interface lemma
.147
12.2
Near the boundary
.152
12.3
Percolation setting
.153
12.4
Lower bound
.155
Part VI Basic geometric tools
13
Sets of finite perimeter
.159
13.1
Basic definitions
.159
13.2
Covering and differentiating
.160
13.3
Caccioppoli sets
.164
13.4
Two technical results
.167
14
Surface energy
.173
14.1
Definition
.175
14.2
Lowersemicontinuity and compactness
.178
14.3
Covering
.178
14.4
Polyhedral approximation
.181
15
The
Wulff
theorem
.189
15.1
Statement of the theorem
.189
15.2
The anisotropic isoperimetric inequality
.190
15.3
The proof of Brothers and Morgan
.192
15.4
Stability of the
Wulff
crystal
.197
Part
VII
Final steps of the proofs
16
LDP for the cluster shapes
.203
16.1
Coarse grained image
.204
16.2
Exponential contiguity
.206
16.3
Local upper bound
.209
16.4
Lower bound
.211
XIV Contents
17
Enhanced upper bound
.215
17.1
A lemma from discrete geometry
.215
17.2
Uniform large deviation upper bounds
.216
17.3
Conclusion of the proof
.221
17.4
Extension to FK percolation
.227
18
LDP for FK percolation
.229
18.1
Coarse grained image
.229
18.2
Exponential contiguity
.233
18.3
Local upper bound
.235
18.4
Lower bound
.236
19
LDP for Ising
.241
19.1
Coarse grained image
.241
19.2
Exponential contiguity
.243
19.3
Local upper bound
.246
19.4
Lower bound
.249
References
.253
Index
.259
List of participants
.261
List of short lectures
.263 |
| adam_txt |
Contents
Part I Introduction
1
Phase coexistence and subadditivity
. 3
1.1
Water and oil
. 3
1.2
Subadditivity
. 5
1.3
Cramer's theorem
. 6
Part II Presentation of the models
2
Ising model
. 15
2.1
Construction of the model
. 15
2.2
First asymptotics
. 16
2.3
Phase transition
. 18
2.4
Proofs of the heuristics
. 18
3
Bernoulli percolation
. 25
3.1
The probability space
. 25
3.2
Order
onß. 26
3.3
Phase transition
. 26
4
FK or random cluster model
. 31
4.1
Finite volume FK measures
. 31
4.2
Phase transition
. 32
4.3
FK Ising coupling
. 33
4.4
Boundary conditions
. 41
XII Contents
Part III Main results
The
Wulff
crystal
. 45
5.1
Ising model
. 45
5.2
Bernoulli percolation
. 55
5.3
FK percolation
. 58
5.4
What do we know about the
Wulff
crystal?
. 60
5.5
Bibliographical comments
. 62
Part IV Large deviation principles
6
Large deviation theory
. 67
6.1
Main definitions
. 67
6.2
/-tightness
. 68
6.3
Contraction principle
. 72
6.4
Varadhan's lemma
. 73
7
Surface large deviation principles
. 75
7.1
Surface energy
. 75
7.2
The empirical magnetization
. 78
7.3
Minimal surfaces
. 79
7.4
The cluster shapes
. 82
7.5
FK percolation
. 83
8
Volume large deviations
. 85
8.1
Bernoulli percolation
. 85
8.2
FK percolation
. 92
8.3
Ising model
. 96
Part V Fundamental probabilistic estimates
9
Coarse graining
.105
9.1
The good blocks
.105
9.2
Extension to FK measures
.110
9.3
The rescaled lattice
.112
9.4
Two rough estimates
.114
10
Decoupling
.117
10.1
Half-space clusters
.117
10.2
Decoupling lemma
.121
Contents XIII
11
Surface tension
.129
11.1
Existence
.129
11.2
Finite
volume
definition
.133
11.3
Basic
properties
.136
11.4
Separating sets
.141
11.5
What do we know about the surface tension?
.145
12
Interface estimate
.147
12.1
Interface lemma
.147
12.2
Near the boundary
.152
12.3
Percolation setting
.153
12.4
Lower bound
.155
Part VI Basic geometric tools
13
Sets of finite perimeter
.159
13.1
Basic definitions
.159
13.2
Covering and differentiating
.160
13.3
Caccioppoli sets
.164
13.4
Two technical results
.167
14
Surface energy
.173
14.1
Definition
.175
14.2
Lowersemicontinuity and compactness
.178
14.3
Covering
.178
14.4
Polyhedral approximation
.181
15
The
Wulff
theorem
.189
15.1
Statement of the theorem
.189
15.2
The anisotropic isoperimetric inequality
.190
15.3
The proof of Brothers and Morgan
.192
15.4
Stability of the
Wulff
crystal
.197
Part
VII
Final steps of the proofs
16
LDP for the cluster shapes
.203
16.1
Coarse grained image
.204
16.2
Exponential contiguity
.206
16.3
Local upper bound
.209
16.4
Lower bound
.211
XIV Contents
17
Enhanced upper bound
.215
17.1
A lemma from discrete geometry
.215
17.2
Uniform large deviation upper bounds
.216
17.3
Conclusion of the proof
.221
17.4
Extension to FK percolation
.227
18
LDP for FK percolation
.229
18.1
Coarse grained image
.229
18.2
Exponential contiguity
.233
18.3
Local upper bound
.235
18.4
Lower bound
.236
19
LDP for Ising
.241
19.1
Coarse grained image
.241
19.2
Exponential contiguity
.243
19.3
Local upper bound
.246
19.4
Lower bound
.249
References
.253
Index
.259
List of participants
.261
List of short lectures
.263 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Cerf, Raphaël |
| author_GND | (DE-588)1044770651 |
| author_facet | Cerf, Raphaël |
| author_role | aut |
| author_sort | Cerf, Raphaël |
| author_variant | r c rc |
| building | Verbundindex |
| bvnumber | BV021608549 |
| callnumber-first | Q - Science |
| callnumber-label | QA3 |
| callnumber-raw | QA3 QC174.85.W85 QC175.16.P5 QE364 |
| callnumber-search | QA3 QC174.85.W85 QC175.16.P5 QE364 |
| callnumber-sort | QA 13 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SI 850 |
| classification_tum | MAT 627f PHY 641f PHY 062f |
| ctrlnum | (OCoLC)70104834 (DE-599)BVBBV021608549 |
| 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 |
| discipline_str_mv | Physik Mathematik |
| format | Book |
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| genre | (DE-588)1071861417 Konferenzschrift gnd-content |
| genre_facet | Konferenzschrift |
| id | DE-604.BV021608549 |
| illustrated | Illustrated |
| index_date | 2024-07-02T14:49:44Z |
| indexdate | 2024-07-20T03:23:34Z |
| institution | BVB |
| isbn | 3540309888 9783540309888 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014823797 |
| oclc_num | 70104834 |
| open_access_boolean | |
| owner | DE-824 DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-83 DE-11 DE-188 |
| owner_facet | DE-824 DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-83 DE-11 DE-188 |
| physical | XIV, 264 S. Ill., graph. Darst. 235 mm x 155 mm |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Cerf, Raphaël Verfasser (DE-588)1044770651 aut The Wulff crystal in ising and percolation models Ecole d'Eté de Probabilités de Saint-Flour XXXIV - 2004 Raphaël Cerf Berlin [u.a.] Springer 2006 XIV, 264 S. Ill., graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1878 Ising-model gtt Percolatietheorie gtt Transformations de phase (Physique statistique) Ising model Percolation (Statistical physics) Phase transformations (Statistical physics) Wulff construction (Statistical physics) Wulff-Konstruktion (DE-588)4440925-4 gnd rswk-swf Perkolationstheorie (DE-588)4323583-9 gnd rswk-swf Ising-Modell (DE-588)4127615-2 gnd rswk-swf Strukturbildung (DE-588)4541700-3 gnd rswk-swf Mehrphasensystem (DE-588)4125888-5 gnd rswk-swf (DE-588)1071861417 Konferenzschrift gnd-content Mehrphasensystem (DE-588)4125888-5 s Strukturbildung (DE-588)4541700-3 s Perkolationstheorie (DE-588)4323583-9 s Ising-Modell (DE-588)4127615-2 s Wulff-Konstruktion (DE-588)4440925-4 s DE-604 Lecture notes in mathematics 1878 (DE-604)BV000676446 1878 Digitalisierung TU Muenchen application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014823797&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cerf, Raphaël The Wulff crystal in ising and percolation models Ecole d'Eté de Probabilités de Saint-Flour XXXIV - 2004 Lecture notes in mathematics Ising-model gtt Percolatietheorie gtt Transformations de phase (Physique statistique) Ising model Percolation (Statistical physics) Phase transformations (Statistical physics) Wulff construction (Statistical physics) Wulff-Konstruktion (DE-588)4440925-4 gnd Perkolationstheorie (DE-588)4323583-9 gnd Ising-Modell (DE-588)4127615-2 gnd Strukturbildung (DE-588)4541700-3 gnd Mehrphasensystem (DE-588)4125888-5 gnd |
| subject_GND | (DE-588)4440925-4 (DE-588)4323583-9 (DE-588)4127615-2 (DE-588)4541700-3 (DE-588)4125888-5 (DE-588)1071861417 |
| title | The Wulff crystal in ising and percolation models Ecole d'Eté de Probabilités de Saint-Flour XXXIV - 2004 |
| title_auth | The Wulff crystal in ising and percolation models Ecole d'Eté de Probabilités de Saint-Flour XXXIV - 2004 |
| title_exact_search | The Wulff crystal in ising and percolation models Ecole d'Eté de Probabilités de Saint-Flour XXXIV - 2004 |
| title_exact_search_txtP | The Wulff crystal in ising and percolation models Ecole d'Eté de Probabilités de Saint-Flour XXXIV - 2004 |
| title_full | The Wulff crystal in ising and percolation models Ecole d'Eté de Probabilités de Saint-Flour XXXIV - 2004 Raphaël Cerf |
| title_fullStr | The Wulff crystal in ising and percolation models Ecole d'Eté de Probabilités de Saint-Flour XXXIV - 2004 Raphaël Cerf |
| title_full_unstemmed | The Wulff crystal in ising and percolation models Ecole d'Eté de Probabilités de Saint-Flour XXXIV - 2004 Raphaël Cerf |
| title_short | The Wulff crystal in ising and percolation models |
| title_sort | the wulff crystal in ising and percolation models ecole d ete de probabilites de saint flour xxxiv 2004 |
| title_sub | Ecole d'Eté de Probabilités de Saint-Flour XXXIV - 2004 |
| topic | Ising-model gtt Percolatietheorie gtt Transformations de phase (Physique statistique) Ising model Percolation (Statistical physics) Phase transformations (Statistical physics) Wulff construction (Statistical physics) Wulff-Konstruktion (DE-588)4440925-4 gnd Perkolationstheorie (DE-588)4323583-9 gnd Ising-Modell (DE-588)4127615-2 gnd Strukturbildung (DE-588)4541700-3 gnd Mehrphasensystem (DE-588)4125888-5 gnd |
| topic_facet | Ising-model Percolatietheorie Transformations de phase (Physique statistique) Ising model Percolation (Statistical physics) Phase transformations (Statistical physics) Wulff construction (Statistical physics) Wulff-Konstruktion Perkolationstheorie Ising-Modell Strukturbildung Mehrphasensystem Konferenzschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014823797&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT cerfraphael thewulffcrystalinisingandpercolationmodelsecoledetedeprobabilitesdesaintflourxxxiv2004 |