Random polymers: École d'Été de Probabilités de Saint-Flour XXXVII - 2007
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Tagungsbericht Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2009
|
| Schriftenreihe: | Lecture notes in mathematics
1974 |
| Schlagworte: | |
| Online-Zugang: | Cover Inhaltsverzeichnis |
| Beschreibung: | XIII, 258 S. Ill., graph. Darst. 235 mm x 155 mm |
| ISBN: | 9783642003325 |
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| 100 | 1 | |a Hollander, Frank den |d 1956- |e Verfasser |0 (DE-588)111877741 |4 aut | |
| 245 | 1 | 0 | |a Random polymers |b École d'Été de Probabilités de Saint-Flour XXXVII - 2007 |c Frank den Hollander |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2009 | |
| 300 | |a XIII, 258 S. |b Ill., graph. Darst. |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Lecture notes in mathematics |v 1974 | |
| 650 | 4 | |a Copolymères - Congrès | |
| 650 | 4 | |a Polymères - Modèles mathématiques - Congrès | |
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Copolymers |v Congresses | |
| 650 | 4 | |a Polymers |x Mathematical models |v Congresses | |
| 650 | 0 | 7 | |a Kettenmolekül |0 (DE-588)4163697-1 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| _version_ | 1804139197172285440 |
|---|---|
| adam_text | Contents
1
Introduction
............................................. 1
1.1
What is a Polymer?
.................................. 1
1.2
What is the Model Setting?
............................ 3
1.3
The Central Role of Free Energy
....................... 6
2
Two Basic Models
....................................... 9
2.1
Simple Random Walk
................................. 9
2.2
Self-avoiding Walk
.................................... 12
Part A Polymers with Self-interaction
3
Soft Polymers in Low Dimension
........................ 19
3.1
A Polymer with Self-repellence
......................... 19
3.2
Weakly Self-avoiding Walk in Dimension One
............ 20
3.3
The Large Deviation Principle for Bridges
............... 22
3.4
Program of Five Steps
................................ 25
3.4.1
Step
1:
Adding Drift
............................ 25
3.4.2
Step
2:
Markovian Nature of the Total Local Times
. 26
3.4.3
Step
3:
Key Variational Problem
................. 27
3.4.4
Step
4:
Solution of the Variational Problem
in Terms of an Eigenvalue Problem
............... 30
3.4.5
Step
5:
Identification of the Speed
................ 33
3.5
The Large Deviation Principle without the Bridge Condition
34
3.6
Extensions
........................................... 35
3.7
Challenges
........................................... 37
4
Soft Polymers in High Dimension
........................ 41
4.1
Weakly Self-avoiding Walk in Dimension Five or Higher
... 41
4.2
Expansion
........................................... 43
4.2.1
Graphs and Connected Graphs
................... 43
4.2.2
Recursion Relation
............................. 45
IX
Contents
4.3
Laces
............................................... 46
4.3.1
Laces and Compatible
Edges
..................... 46
4.3.2
Resummation
.................................. 48
4.4
Diagrammatic Estimates
.............................. 49
4.5
Induction
............................................ 51
4.5.1
Notation
...................................... 51
4.5.2
Heuristics
..................................... 52
4.5.3
Induction Hypotheses
........................... 53
4.6
Proof of Diffusive Behavior
............................ 54
4.7
Extensions
........................................... 56
4.8
Challenges
........................................... 58
Elastic Polymers
......................................... 59
5.1
A Polymer with Decaying Self-repellence
................ 59
5.2
The Lace Expansion Carries over
....................... 60
5.3
Extensions
........................................... 61
5.4
Challenges
........................................... 63
Polymer Collapse
........................................ 67
6.1
An Undirected Polymer in a Poor Solvent
............... 67
6.1.1
The Minimally Extended Phase
.................. 69
6.1.2
The Localized Phase
............................ 70
6.1.3
Conjectured Phase Diagram
..................... 73
6.2
A Directed Polymer in a Poor Solvent
................... 74
6.2.1
The Collapse Transition
......................... 76
6.2.2
Properties of the Two Phases
.................... 80
6.3
Extensions
........................................... 81
6.4
Challenges
........................................... 83
Polymer Adsorption
..................................... 85
7.1
A Polymer Near a Linear Penetrable Substrate: Pinning
... 86
7.1.1
Free Energy
................................... 88
7.1.2
Path Properties
................................ 91
7.1.3
Order of the Phase Transition
.................... 93
7.2
A Polymer Near a Linear Impenetrable Substrate: Wetting
93
7.3
Pulling a Polymer off a Substrate by a Force
............. 95
7.3.1
Force and Pinning
.............................. 96
7.3.2
Force and Wetting
.............................. 99
7.4
A Polymer in a Slit between Two Impenetrable Substrates
. 101
7.4.1
Model
........................................101
7.4.2
Generating Function
............................103
7.4.3
Effective Force
.................................104
7.5
Adsorption of Self-avoiding Walks
......................106
7.6
Extensions
...........................................107
7.7
Challenges
...........................................
Ш
Contents
XI
Part
В
Polymers in Random Environment
8
Charged Polymers
.......................................115
8.1
A Polymer with Screened Random Charges
..............116
8.2
Scaling of the Free Energy
.............................117
8.2.1
Variational Characterization
.....................117
8.2.2
Heuristics
.....................................119
8.2.3
Large Deviations
...............................121
8.3
Subdiffusive
Behavior
.................................121
8.4
Parabolic Anderson Equation
..........................122
8.5
Extensions
...........................................123
8.6
Challenges
...........................................126
9
Copolymers Near a Linear Selective Interface
............129
9.1
A Copolymer Interacting with Two Solvents
.............130
9.2
The Free Energy
.....................................132
9.3
The Critical Curve
...................................135
9.3.1
The Localized and Delocalized Phases
............135
9.3.2
Existence of a Non-trivial Critical Curve
..........136
9.4
Qualitative Properties of the Critical Curve
..............138
9.4.1
Upper Bound
..................................138
9.4.2
Lower Bound
..................................138
9.4.3
Weak Interaction Limit
.........................141
9.5
Qualitative Properties of the Phases
....................142
9.5.1
Path Properties
................................143
9.5.2
Order of the Phase Transition
....................143
9.5.3
Smoothness of the Free Energy
in the Localized Phase
..........................145
9.6
Extensions
...........................................147
9.7
Challenges
...........................................153
10
Copolymers Near a Random Selective Interface
..........155
10.1
A Copolymer Diagonally Crossing Blocks
................156
10.2
Preparations
.........................................158
10.2.1
Variational Formula for the Free Energy
...........158
10.2.2
Path Entropies
.................................161
10.2.3
Free Energies per Pair of Blocks
..................162
10.2.4
Percolation
....................................164
10.3
Phase Diagram in the Supercritical Regime
..............165
10.3.1
Free Energy in the Two Phases
..................166
10.3.2
Criterion for Localization
........................167
10.3.3
Qualitative Properties of the Critical Curve
........169
10.3.4
Finer Details of the Critical Curve
................173
XII Contents
10.4 Phase Diagram in
the
Subcriticai Regime
................175
10.5
Extensions...........................................
177
10.6
Challenges
...........................................178
11
Random Pinning and Wetting of Polymers
..............181
11.1
A Polymer Near a Linear Penetrable Random Substrate:
Pinning
.............................................182
11.2
The Free Energy
.....................................183
11.3
The Critical Curve
...................................184
11.3.1
The Localized and Delocalized Phases
............184
11.3.2
Existence of a Non-trivial Critical Curve
..........184
11.4
Qualitative Properties of the Critical Curve
..............185
11.4.1
Upper Bound
..................................185
11.4.2
Lower Bound
..................................186
11.4.3
Weak Interaction Limit
.........................190
11.5
Qualitative Properties of the Phases
....................191
11.6
Relevant versus Irrelevant Disorder
.....................191
11.7
A Polymer Near a Linear Impenetrable Random
Substrate: Wetting
...................................194
11.8
Pulling a Polymer off a Substrate by a Force
.............195
11.8.1
Force and Pinning
..............................196
11.8.2
Force and Wetting
..............................197
11.9
Extensions
...........................................198
11.10
Challenges
...........................................204
12
Polymers in a Random Potential
........................205
12.1
A Homopolymer in a Micro-emulsion
...................206
12.2
A Dichotomy: Weak and Strong Disorder
................207
12.2.1
Key Martingale
................................207
12.2.2
Separation of the Two Phases
....................209
12.2.3
Characterization of the Two Phases
..............209
12.2.4
Diffusive versus Non-diffusive Behavior
............211
12.2.5
Bounds on the Critical Temperature
..............211
12.3
Proof of Uniqueness of the Critical Temperature
..........212
12.4
Martingale Estimates
.................................214
12.4.1
First Estimate
.................................215
12.4.2
Second Estimate
...............................218
12.5
The Weak Disorder Phase
.............................220
12.6
The Strong Disorder Phase
............................220
12.7
Beyond Second Moments
..............................221
12.7.1
Fractional Moment Estimates
....................222
12.7.2
Size-biasing
....................................223
12.7.3
Relation with Random Pinning
..................225
Contents XIII
12.8
Extensions
...........................................226
12.9
Challenges
...........................................230
References.....................................................
233
Index
..........................................................249
List of
Participants
............................................253
Programme
of the School
......................................257
|
| any_adam_object | 1 |
| author | Hollander, Frank den 1956- |
| author_GND | (DE-588)111877741 |
| author_facet | Hollander, Frank den 1956- |
| author_role | aut |
| author_sort | Hollander, Frank den 1956- |
| author_variant | f d h fd fdh |
| building | Verbundindex |
| bvnumber | BV035551905 |
| callnumber-first | Q - Science |
| callnumber-label | QD381 |
| callnumber-raw | QD381.9.M3 |
| callnumber-search | QD381.9.M3 |
| callnumber-sort | QD 3381.9 M3 |
| callnumber-subject | QD - Chemistry |
| classification_rvk | SI 850 |
| classification_tum | PHY 057f PHY 624f MAT 609f |
| ctrlnum | (OCoLC)318871642 (DE-599)DNB994021879 |
| 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 |
| format | Conference Proceeding Book |
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| indexdate | 2024-07-09T21:40:15Z |
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| isbn | 9783642003325 |
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| series2 | Lecture notes in mathematics |
| spelling | Hollander, Frank den 1956- Verfasser (DE-588)111877741 aut Random polymers École d'Été de Probabilités de Saint-Flour XXXVII - 2007 Frank den Hollander Berlin [u.a.] Springer 2009 XIII, 258 S. Ill., graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1974 Copolymères - Congrès Polymères - Modèles mathématiques - Congrès Mathematisches Modell Copolymers Congresses Polymers Mathematical models Congresses Kettenmolekül (DE-588)4163697-1 gnd rswk-swf Stochastisches Modell (DE-588)4057633-4 gnd rswk-swf (DE-588)1071861417 Konferenzschrift gnd-content Kettenmolekül (DE-588)4163697-1 s Stochastisches Modell (DE-588)4057633-4 s DE-604 Ecole d'Eté de Probabilités 37 2007 Saint-Flour Sonstige (DE-588)16178642-X oth Erscheint auch als Online-Ausgabe 978-3-642-00333-2 Lecture notes in mathematics 1974 (DE-604)BV000676446 1974 DE-576;springer image/jpeg http://swbplus.bsz-bw.de/bsz307031667cov.htm 20090528182920 Cover Digitalisierung TU Muenchen application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017607794&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hollander, Frank den 1956- Random polymers École d'Été de Probabilités de Saint-Flour XXXVII - 2007 Lecture notes in mathematics Copolymères - Congrès Polymères - Modèles mathématiques - Congrès Mathematisches Modell Copolymers Congresses Polymers Mathematical models Congresses Kettenmolekül (DE-588)4163697-1 gnd Stochastisches Modell (DE-588)4057633-4 gnd |
| subject_GND | (DE-588)4163697-1 (DE-588)4057633-4 (DE-588)1071861417 |
| title | Random polymers École d'Été de Probabilités de Saint-Flour XXXVII - 2007 |
| title_auth | Random polymers École d'Été de Probabilités de Saint-Flour XXXVII - 2007 |
| title_exact_search | Random polymers École d'Été de Probabilités de Saint-Flour XXXVII - 2007 |
| title_full | Random polymers École d'Été de Probabilités de Saint-Flour XXXVII - 2007 Frank den Hollander |
| title_fullStr | Random polymers École d'Été de Probabilités de Saint-Flour XXXVII - 2007 Frank den Hollander |
| title_full_unstemmed | Random polymers École d'Été de Probabilités de Saint-Flour XXXVII - 2007 Frank den Hollander |
| title_short | Random polymers |
| title_sort | random polymers ecole d ete de probabilites de saint flour xxxvii 2007 |
| title_sub | École d'Été de Probabilités de Saint-Flour XXXVII - 2007 |
| topic | Copolymères - Congrès Polymères - Modèles mathématiques - Congrès Mathematisches Modell Copolymers Congresses Polymers Mathematical models Congresses Kettenmolekül (DE-588)4163697-1 gnd Stochastisches Modell (DE-588)4057633-4 gnd |
| topic_facet | Copolymères - Congrès Polymères - Modèles mathématiques - Congrès Mathematisches Modell Copolymers Congresses Polymers Mathematical models Congresses Kettenmolekül Stochastisches Modell Konferenzschrift |
| url | http://swbplus.bsz-bw.de/bsz307031667cov.htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017607794&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT hollanderfrankden randompolymersecoledetedeprobabilitesdesaintflourxxxvii2007 AT ecoledetedeprobabilitessaintflour randompolymersecoledetedeprobabilitesdesaintflourxxxvii2007 |