Statistical mechanics:
Gespeichert in:
| Weitere Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, RI
American Math. Soc.
2009
|
| Schriftenreihe: | IAS - Park City mathematics series
16 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturangaben |
| Beschreibung: | XII, 360 S. graph. Darst. |
| ISBN: | 9780821846711 |
Internformat
MARC
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| 245 | 1 | 0 | |a Statistical mechanics |c Scott Sheffield ... eds. |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2009 | |
| 300 | |a XII, 360 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a IAS - Park City mathematics series |v 16 | |
| 500 | |a Literaturangaben | ||
| 650 | 4 | |a Statistical mechanics | |
| 650 | 0 | 7 | |a Statistische Mechanik |0 (DE-588)4056999-8 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)1071861417 |a Konferenzschrift |2 gnd-content | |
| 689 | 0 | 0 | |a Statistische Mechanik |0 (DE-588)4056999-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Sheffield, Scott |4 edt | |
| 830 | 0 | |a IAS - Park City mathematics series |v 16 |w (DE-604)BV010402400 |9 16 | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-018600770 | |
Datensatz im Suchindex
| _version_ | 1820889684369211392 |
|---|---|
| adam_text |
Contents
Preface
xi
Scott Sheffield and Thomas Spencer
Introduction
1
David
С
Brydges
Lectures on
the
Renormalisation Group
7
Acknowledgment
9
Lecture
1.
Scaling limits and Gaussian measures
11
1.1.
Introduction
11
1.2.
Theoretical physics
13
1.3.
Some results
15
1.4.
Gaussian measures on
Мл
15
1.5.
Example: One dimension
18
1.6.
Local and global functions
19
1.7.
Example: Particles on a lattice
21
1.8.
The importance of the partition function
22
1.9.
Appendix. Green's functions
24
Lecture
2.
Renormalisation
group in hierarchical models
27
2.1.
Massless Gaussian measure
27
2.2.
Finite range decompositions
27
2.3.
Motivation
29
2.4.
The
renormalisation
group and hierarchical models
30
2.5.
Hierarchical models
31
2.6.
The formal infinite volume limit and trivial fixed point
34
2.7.
Analysis
35
2.8.
Expanding directions, relevant operators
35
2.9.
Trivial fixed point
36
2.10.
Appendix. Stable manifold theorem
37
Lecture
3.
Example: the hierarchical Coulomb gas
41
3.1.
Example: Hierarchical Coulomb gas
41
3.2.
Finite volume
46
3.3.
Fractional charge observable and confinement
47
3.4.
Appendix. Notes on the rigorous
renormalisation
group
49
Lecture
4.
Renormalisation
group for Euclidean models
53
4.1.
Euclidean lattice and the
dipole
model
53
4.2.
The initial
/ο,ΛΌ
54
4.3.
The basic scaling mechanism
55
4.4.
Coordinates
(/j.Äj·) 55
4.5.
Euclidean replacement for Lemma
2.14 57
vi
CONTENTS
Lecture
5.
Coordinates and action of
renormalisation
group
61
5.1.
Euclidean replacement for Lemma
2.14
continued
61
5.2.
Formulas for
ï, J.
64
Lecture
6.
Smoothness of (RG)
67
6.1.
Choice of spaces and smoothness of (RG)
67
6.2.
Norms
71
6.3.
Open problems
78
6.4.
Appendix. Geometry and counting lemmas
78
6.5.
Appendix. Proof of Theorem
6.14 82
Bibliography
91
Alice Guionnet
Statistical Mechanics and Random Matrices
95
Introduction
97
1.
Motivations
98
2.
The different scales; typical results
104
Lecture
1.
Wigner matrices and moments estimates
109
1.
Wigner's theorem
109
2.
Words in several independent Wigner matrices
118
3.
Estimates on the largest eigenvalue of Wigner matrices
120
Lecture
2.
Gaussian Wigner matrices and
Fredholm
determinants
123
1.
Joint law of the eigenvalues
123
2.
Joint law of the eigenvalues and
determinanta!
law
124
3.
Determinantal structure and Fredholm determinants
126
4.
Fredholm determinant and asymptotics
127
Lecture
3.
Wigner matrices and concentration inequalities
131
1.
Concentration inequalities and logarithmic Sobolev inequalities
131
2.
Smoothness and convexity of the eigenvalues of a matrix and of traces
of matrices
135
3.
Concentration inequalities for random matrices
139
4.
Brascamp-Lieb inequalities; applications to random matrices
141
Lecture
4.
Matrix models
147
1.
Combinatorics of maps and non-commutative polynomials
149
2.
Formal expansion of matrix integrals
153
3.
First order expansion for the free energy
160
4.
Discussion
167
Lecture
5.
Random matrices and dynamics
171
1.
Free Brownian motions and related stochastic differential calculus
172
2.
Consequences
179
3.
Discussion
182
Bibliography
185
CONTENTS
vii
Richard Kenyon
Lectures on
Dimers
191
193
193
194
196
198
199
200
200
200
201
202
203
203
203
204
205
206
206
207
209
209
209
209
210
210
211
211
6. Uniform
honeycomb dimers
212
6.1.
Inverse Kasteleyn matrix
213
6.2.
Decay of correlations
213
6.3.
Height fluctuations
214
7.
Legendre duality
215
8.
Boundary conditions
217
9.
Burgers equation
218
9.1.
Volume constraint
220
9.2.
Frozen boundary
220
9.3.
General solution
220
10.
Amoebas and Harnack curves
222
10.1.
The amoeba of
Ρ
222
10.2.
Phases of EGMs
223
1.
Overview
1.1.
Dimer definitions
1.2.
Uniform random tilings
1.3.
Limit shapes
1.4.
Facets
1.5.
Measures
1.6.
Other random surface models
2.
Th(
з
height function
2.1.
Graph homology
2.2.
Heights
3.
Kasteleyn theory
3.1.
The Boltzmann measure
3.2.
Gauge equivalence
3.3.
Kasteleyn weighting
3.4.
Kasteleyn matrix
3.5.
Local statistics
4.
Partition function
4.1.
Rectangle
4.2.
Torus
4.3.
Partition function
4.4.
Height change distribution
5.
Gibbs measures
5.1.
Definition
5.2.
Periodic graphs
5.3.
Ergodic Gibbs measures
5.4.
Constructing EGMs
5.5.
Magnetic field
viii CONTENTS
10.3. Harnack
curves
224
10.4.
Example
224
11.
Fluctuations
226
11.1.
The Gaussian free field
226
11.2.
On the plane
227
11.3.
Gaussians and moments
227
11.4.
Height fluctuations on the plane
227
12.
Open problems
229
Bibliography
229
G. Lawler
Schramm-Loewner Evolution (SLE)
231
Introduction
233
Lecture
1.
Scaling limits of lattice models
235
1.
Self-avoiding walk (SAW)
235
2.
Loop-erased random walk
239
3.
Percolation
241
4.
Ising model
242
5.
Assumptions on limits
243
6.
Exercises for Lecture
1 243
Lecture
2.
Conformai
mapping and Loewner equation
245
1.
Important results about
conformai
maps
245
2.
Half-plane capacity
247
3.
Loewner equation
249
4.
Maps generated by a curve
251
5.
A flow on
conformai
maps
252
6.
Doubly infinite time
253
7.
Distance to boundary
254
8.
Exercises for Lecture
2 255
Lecture
3.
Schramm-Loewner evolution (SLE)
257
1.
Definition
257
2.
Phases
259
3.
Dimension of the path
260
4.
Cardy's formula
261
5.
Conformai
images of SLE
262
6.
Exercises for Lecture
3 264
Lecture
4.
SLEK in a simply connected domain
D
265
1.
Drift and locality
265
2.
Girsanov
266
3.
The restriction martingale
267
4.
(Brownian) boundary bubbles
268
5.
Brownian loop measure
270
6.
The measure
μο(*,
w)
for
к <
4 271
7.
Exercises for Lecture
4 272
CONTENTS ix
Lecture
5. Radial
and two-sided radial SLBK 27b
1.
Example: SAW II
275
2.
Radial SLEK
277
3.
Another definition
279
4.
Radial SLEK in a smaller domain
280
5.
Two-sided radial
281
6.
Exercises for Lecture
5 282
Lecture
6.
Intersection exponents
283
1.
One-sided
283
2.
Two-sided
286
3.
Nonintersecting SLEK
287
4.
Radial exponent and SAW III
288
5.
Exercises for Lecture
6 290
Tables for reference
291
Bibliography
293
Wendelin
Werner
Lectures on Two-dimensional Critical Percolation
297
Overview
299
Lecture
1.
Introduction and tightness
301
1.
2D percolation
301
2.
Notations and prerequisites
302
3.
Russo-Seymour-Welsh
304
4.
First consequences
306
First exercise sheet
309
Lecture
2.
The Cardy-Smirnov formula
313
1.
Preliminaries
313
2.
Smirnov's theorem
315
Lecture
3.
Convergence to SLE(6)
321
1.
Our goal
321
2.
Hand-waving argument
322
3.
Tightness
323
4.
Loewner chains
323
5.
Capacity increases
324
6.
Side-remark concerning the proof of Cardy-Smirnov formula
326
7.
Identifying continuous martingales
326
8.
Recognizing SLE(6)
327
9.
Take-home message
328
Second exercise sheet
331
Lecture
4.
SLE(6) computations
333
1.
Radial SLE
333
2.
Relation between radial and chordal SLE(6)
334
χ
CONTENTS
3.
Relation to discrete radial exploration
335
4.
Exponent computations
336
Lecture
5.
Arm exponents
341
1.
Some notations
341
2.
One-arm exponent
341
3.
Four-arms exponent
343
4.
Other exponents and bibliographical remarks
349
Lecture
6.
Near-critical percolation
351
1.
Correlation length
351
2.
Outline of the proof
352
3.
A priori estimates
353
4.
Arm separation
353
5.
Using differential inequalities
354
6.
Concluding remarks
358
Bibliography
359 |
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| indexdate | 2025-01-10T19:02:04Z |
| institution | BVB |
| isbn | 9780821846711 |
| language | English |
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| spelling | Statistical mechanics Scott Sheffield ... eds. Providence, RI American Math. Soc. 2009 XII, 360 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier IAS - Park City mathematics series 16 Literaturangaben Statistical mechanics Statistische Mechanik (DE-588)4056999-8 gnd rswk-swf (DE-588)1071861417 Konferenzschrift gnd-content Statistische Mechanik (DE-588)4056999-8 s DE-604 Sheffield, Scott edt IAS - Park City mathematics series 16 (DE-604)BV010402400 16 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018600770&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Statistical mechanics IAS - Park City mathematics series Statistical mechanics Statistische Mechanik (DE-588)4056999-8 gnd |
| subject_GND | (DE-588)4056999-8 (DE-588)1071861417 |
| title | Statistical mechanics |
| title_auth | Statistical mechanics |
| title_exact_search | Statistical mechanics |
| title_full | Statistical mechanics Scott Sheffield ... eds. |
| title_fullStr | Statistical mechanics Scott Sheffield ... eds. |
| title_full_unstemmed | Statistical mechanics Scott Sheffield ... eds. |
| title_short | Statistical mechanics |
| title_sort | statistical mechanics |
| topic | Statistical mechanics Statistische Mechanik (DE-588)4056999-8 gnd |
| topic_facet | Statistical mechanics Statistische Mechanik Konferenzschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018600770&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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