Mathematical Statistical Physics

, by Bovier; Dunlop; Van Enter; Den Hollander; Dalibard

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ISBN: 9780444528131 | 044452813X
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Copyright: 8/24/2006

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The proceedings of the 2005 les Houches summer school on Mathematical Statistical Physics give and broad and clear overview on this fast developing area of interest to both physicists and mathematicians. · introduction to a field of math with many interdisciplinary connections in physics, biology, and computer science · roadmap to the next decade of mathematical statistical mechanics · volume for reference years to come

Organizers

Lecturers

Participants

Preface

Informal seminars

xix

Course 1. Random matrices and determinantal processes,

by Kurt Johansson

(56)

1. Introduction

(1)

2. Point processes

(18)

2.1. General theory

(8)

2.2. Determinantal processes

(6)

2.3. Measures defined by products of several determinants

(4)

3. Non-intersecting paths and the Aztec diamond

(10)

3.1. Non-intersecting paths and the LGV theorem

(2)

3.2. The Aztec diamond

(5)

3.3. Relations to other models

(3)

4. Asymptotics

(10)

4.1. Double contour integral formula for the correlation kernel

(2)

4.2. Asymptotics for the Aztec diamond

(6)

4.3. Asymptotics for random permutations

(2)

5. The corner growth model

(10)

5.1. Mapping to non-intersecting paths

(2)

5.2. The Schur and Plancherel measures

(3)

5.3. A discrete polynuclear growth model

(2)

5.4. Proof of theorem 5.1

(3)

References

(4)

Course 2. Some recent aspects of random conformally invariant systems,

by Wendelin Werner

(44)

1. Some discrete models

(10)

1.1. Self-avoiding walks and polygons

(1)

1.2. Random walk loops

(1)

1.3. Site-percolation

(2)

1.4. The Ising model

(2)

1.5. The Potts models

(1)

1.6. FK representations of Potts models

(1)

1.7. The O (N) models

(2)

1.8. Conformal invariance

(2)

2. A "conformal Haar measure" on self-avoiding loops

(7)

2.1. Preliminaries

(1)

2.2. A conformal invariance property

(1)

2.3. Uniqueness

(3)

2.4. Existence

(3)

3. Schramm-Loewner Evolutions

(7)

3.1. Definition

(3)

3.2. Computing with SLE

(4)

4. Conformal loop-ensembles

(10)

4.1. Definition

(2)

4.2. First properties

(1)

4.3. The loop-soup construction

(4)

4.4. The Gaussian free field

(3)

4.4.1. Definition

(1)

4.4.2. "Cliffs" as level lines

(2)

References

(4)

Course 3. Conformal random geometry,

by Bertrand Duplantier

101

(4)

1. Preamble

105

(2)

2. Introduction

107

(112)

2.1. A brief conformal history

107

(3)

2.2. Conformal geometrical structures

110

(1)

2.3. Quantum gravity

111

(1)

2.4. Stochastic Löwner evolution

112

(1)

2.5. Recent developments

113

(2)

2.6. Synopsis

115

(2)

3. Intersections of random walks

117

(1)

3.1. Non-intersection probabilities

117

(3)

3.2. Quantum gravity

120

(3)

3.3. Random walks on a random lattice

123

(9)

3.4. Non-intersections of packets of walks

132

(5)

4. Mixing random & self-avoiding walks

137

(1)

4.1. General star configurations

137

(3)

4.2. Quantum gravity for SAW's & RW's

140

(5)

4.3. RW-SAW exponents

145

(1)

4.4. Brownian hiding exponents

146

(2)

5. Percolation clusters

148

(1)

5.1. Cluster hull and external perimeter

148

(2)

5.2. Harmonic measure of percolation frontiers

150

(1)

5.3. Harmonic and path crossing exponents

151

(1)

5.4. Quantum gravity for percolation

152

(1)

5.5. Multifractality of percolation clusters

153

(3)

6. Conformally invariant frontiers and quantum gravity

156

(1)

6.1. Harmonic measure and potential near a fractal frontier

156

(4)

6.2. Calculation of multifractal exponents from quantum gravity

160

(7)

6.3. Geometrical analysis of multifractal spectra

167

(6)

7. Higher multifractal spectra

173

(1)

7.1. Double-sided spectra

173

(5)

7.2. Higher multifractality of multiple path vertices

178

(1)

8. Winding of conformally invariant curves

178

(1)

8.1. Harmonic measure and rotations

179

(1)

8.2. Exact mixed multifractal spectra

180

(3)

8.3. Conformal invariance and quantum gravity

183

(2)

8.4. Rotation scaling exponents

185

(2)

8.5. Legendre transform

187

(1)

9. O(N) & Potts models and the Stochastic Löwner Evolution

187

(1)

9.1. Geometric duality in O(N) and Potts cluster frontiers

187

(3)

9.2. Geometric duality of SLEK

190

(3)

10. Quantum gravity duality and SLE

193

(16)

10.1. Dual dimensions

193

(3)

10.2. KPZ for SLE

196

(2)

10.3. Short distance expansion

198

(2)

10.4. Multiple paths in O(N), Potts models and SLE

200

(2)

10.5. SLE (kappa, ρ) and quantum gravity

202

(2)

10.6. Multifractal exponents for multiple SLE's

204

(5)

References

209

(10)

Course 4. Random motions in random media,

by Alain-Sol Sznitman

219

(24)

1. Introduction

223

(1)

2. RWRE

224

(1)

3. RCM

225

(4)

4. Back to RWRE

229

(6)

5. Diffusions in random environment

235

(2)

References

237

(6)

Course 5. An introduction to mean field spin glas theory: methods and results,

by Francesco Guerra

243

(30)

1. Introduction

247

(4)

2. The mean field ferromagnetic model. Convexity and cavity methods

251

(4)

3. The mean field spin glass model. Basic definitions

255

(2)

4. The interpolation method and its generalizations

257

(3)

5. The thermodynamic limit and the variational bounds

260

(3)

6. The Parisi representation for the free energy

263

(5)

7. Conclusion and outlook for future developments

268

(1)

References

269

(4)

Course 6. Short-range spin glasses: selected open problems,

by C.M. Newman and D.L. Stein

273

(22)

1. Introduction

277

(4)

2. The Fortuin-Kasteleyn random cluster representation and phase transitions

281

(3)

3. Spin glass ground states and invasion percolation

284

(4)

4. Ground state multiplicity in the 2D EA spin glass

288

(2)

References

290

(5)

Course 7. Computing the number of metastable states in infinite-range models,

by Giorgio Parisi

295

(36)

1. Introduction

299

(4)

2. The TAP equations

303

(4)

3. A simple analysis of the solutions of the Bethe equations

307

(5)

4. The direct approach: general considerations

312

(4)

5. The supersymmetric formulation

316

(2)

6. Spontaneous supersymmetry breaking

318

(3)

7. An explicit computation: the complexity of the SK model

321

(3)

8. A few words on quenched complexity

324

(1)

9. Conclusions and open problems

325

(2)

References

327

(4)

Course 8. Dynamics of trap models,

by Gérard Ben Arous and Jiri Cerný

331

(64)

1. Introduction

335

(3)

2. Definition of the Bouchaud trap model

338

(6)

2.1. Examples of trap models

341

(1)

2.2. Natural questions on trap models

342

(1)

2.3. References

343

(1)

3. The one-dimensional trap model

344

(12)

3.1. The Fontes-Isopi-Newman singular diffusion

344

(2)

3.2. The scaling limit

346

(6)

3.2.1. Time-scale change of Brownian motion

347

(2)

3.2.2. Convergence of the fixed-time distributions

349

(1)

3.2.3. A coupling for walks on different scales

349

(2)

3.2.4. Scaling limit

351

(1)

3.3. Aging results

352

(1)

3.4. Subaging results

353

(1)

3.5. Behaviour of the aging functions on different time scales

354

(2)

3.6. References

356

(1)

4. The trap model in dimension larger than one

356

(7)

4.1. The fractional-kinetics process

356

(2)

4.2. Scaling limit

358

(1)

4.3. Aging results

359

(1)

4.4. The coarse-graining procedure

360

(3)

4.5. References

363

(1)

5. The arcsine law as a universal aging scheme

363

(17)

5.1. Aging on large complete graphs

364

(9)

5.1.1. Deep traps

367

(2)

5.1.2. Shallow traps

369

(2)

5.1.3. Very deep traps

371

(1)

5.1.4. Proof of Theorem 5.1

371

(2)

5.2. The α-stable subordinator as a universal clock

373

(4)

5.3. Potential-theoretic characterisation

377

(3)

6. Applications of the arcsine law

380

(7)

6.1. Aging in the REM

380

(6)

6.1.1. Short time scales

381

(2)

6.1.2. Long time scales

383

(2)

6.1.3. Open questions and conjectures

385

(1)

6.2. Aging on large tori

386

(1)

Appendix A. Subordinators

387

(4)

References

391

(4)

Course 9. Quantum entropy and quantum information,

by Nilanjana Datta

395

(72)

1. Introduction

399

(1)

2. Rudiments of Classical Information Theory

400

(17)

2.1. Entropy in Classical Information Theory

405

(4)

2.1.1. Entropy of a pair of random variables

406

(3)

2.2. Shannon's Noiseless Channel Coding Theorem

409

(5)

2.2.1. Asymptotic Equipartition Property (AEP)

410

(2)

2.2.2. Consequences of the AEP

412

(2)

2.3. Information transmission and Channel Capacity

414

(3)

3. Introduction to Quantum Information Theory

417

(3)

4. Open systems

420

(14)

4.0.1. Properties of the density matrix

421

(1)

4.0.2. Reduced density matrix and partial trace

422

(5)

4.1. Time evolution of open systems

427

(3)

4.2. Generalized measurements

430

(1)

4.3. Implementation of a generalized measurement

431

(3)

5. Quantum entropy

434

(7)

5.1. Properties of the von Neumann entropy S(ρ)

436

(5)

6. Data compression in Quantum Information Theory

441

(7)

6.1. Schumacher's Theorem for memoryless quantum sources

442

(6)

7. Quantum channels and additivity

448

(15)

7.1. The noise in the channel

450

(1)

7.2. Capacities of a quantum channel

451

(1)

7.3. Classical capacity of a quantum channel

452

(6)

7.4. A sufficient condition for additivity

458

(4)

7.5. Multiplicativity of the maximal p—norm

462

(1)

Bibliography

463

(4)

Course 10. Two lectures on iterative coding and statistical mechanics,

by Andrea Montanari

467

(22)

1. Introduction

471

(1)

2. Codes on graphs

472

(1)

3. A simple—minded bound and belief propagation

473

(6)

3.1. Characterizing the code performances

473

(1)

3.2. Bounding the conditional entropy

474

(4)

3.3. A parenthesis

478

(1)

4. Density evolution a.k.a. distributional recursive equations

479

(3)

5. The area theorem and some general questions

482

(2)

6. Historical and bibliographical note

484

(1)

References

485

(4)

Course 11. Evolution in fluctuating populations,

by Alison Etheridge

489

(58)

1. Introduction

493

(2)

2. Some classical coalescent theory

495

(3)

2.1. Kingman's coalescent

495

(2)

2.2. Variable population size

497

(1)

2.3. Introducing structure

497

(1)

3. Fluctuations matter

498

(16)

3.1. Balancing selection

499

(3)

3.2. A second neutral locus

502

(1)

3.3. The problem

503

(1)

3.4. The ancestral recombination graph and local trees

504

(2)

3.5. Back to the main plot

506

(1)

3.6. The diffusion approximation

507

(5)

3.7. Extensions

512

(1)

3.8. Summary and notes of caution

513

(1)

4. Spatial structure and the Malécot formula

514

(15)

4.1. Branching process models

514

(3)

4.2. The classical stepping stone model

517

(2)

4.3. Duality

519

(2)

4.4. Neutral evolution

521

(2)

4.5. Random walks in random environments

523

(1)

4.6. Models in continuous space

523

(1)

4.7. Malècot's formula

524

(5)

5. Spatial models

529

(13)

5.1. Locally regulated populations

529

(4)

5.2. Competing species

533

(4)

5.3. Branching annihilating random walk

537

(1)

5.4. A duality and a conjecture for Model II

538

(1)

5.5. Conjectures for Model I

539

(1)

5.6. Heteromyopia

540

(2)

References

542

(5)

Course 12. Multi-scale analysis of population models,

by A. Greven

547

(60)

1. Spatial diffusion models of population genetics

551

(5)

2. Duality and representation via coalescent processes

556

(10)

2.1. The function-valued dual process

556

(4)

2.2. The refined dual process

560

(5)

2.3. Application: ergodic theorem

565

(1)

3. Historical processes

566

(11)

3.1. The neutral models: IMM and IFWD

566

(2)

3.2. The concept of the historical process

568

(1)

3.3. The concept of the look-down process

569

(3)

3.4. Representation theorem via look-down

572

(3)

3.5. Representation via coalescent

575

(2)

4. Quasi-equilibria: elements of the multi-scale analysis

577

(8)

4.1. Mean-field models and McKean-Vlasov limit

578

(1)

4.2. The mean-field dual

579

(2)

4.3. Longtime behavior of McKean-Vlasov process

581

(1)

4.4. Large time-space scale behavior of mean-field model

582

(3)

5. Punctuated equilibria and hierarchical mean-field limit

585

(17)

5.1. Detailed specification of selection and mutation mechanisms

586

(2)

5.2. Interpretation of the model from the point of view of evolutionary theory

588

(1)

5.3. The basic scenario

589

(2)

5.4. The hierarchical mean-field limit

591

(2)

5.5. Two interaction chains

593

(1)

5.6. Basic limit theorem

594

(2)

5.7. Punctuated equilibria: emergence and take-over

596

(6)

References

602

(5)

Course 13. Elements of nonequilibrium statistical mechanics,

by Christian Maes

607

(50)

1. Elements of introduction

611

(2)

1.1. The goal

611

(1)

1.2. The plan

612

(1)

2. Elements of an H-theorem

613

(8)

2.1. Heuristics of an H-theorem

613

(1)

2.2. H-theorem: the problem

614

(1)

2.3. Macroscopic H-theorem

615

(4)

2.3.1. Semigroup property

616

(1)

2.3.2. Autonomy on a finite partition

617

(1)

2.3.3. Reversibility

617

(1)

2.3.4. Propagation of constrained equilibrium

618

(1)

2.4. Pathwise H-theorem

619

(2)

3. Elements of heat conduction

621

(8)

3.1. Hamiltonian dynamics

621

(1)

3.2. Non-uniqueness

622

(1)

3.3. Boundary conditions to the Hamiltonian flow

623

(1)

3.4. Heat bath dynamics

624

(1)

3.5. Entropy production and heat currents

624

(2)

3.6. Mean entropy production

626

(1)

3.7. Strict positivity of mean entropy production

627

(1)

3.8. Getting work done

627

(2)

4. Lagrangian approach

629

(3)

4.1. First goal

629

(1)

4.2. Entropy in irreversible thermodynamics

630

(2)

5. A little entropology

632

(3)

5.1. Reduced variables

632

(1)

5.2. Distributions

633

(1)

5.3. Entropy functional

633

(1)

5.4. Microcanonical entropy

634

(1)

5.5. Equilibrium entropy

634

(1)

5.6. Dynamical entropy

635

(1)

6. Closed systems

635

(3)

6.1. Extensions

636

(1)

6.2. Example: diffusion in local thermodynamic equilibrium

636

(2)

7. Open systems

638

(3)

7.1. Heat conduction

638

(1)

7.2. Asymmetric exclusion process

638

(2)

7.3. Strongly chaotic dynamical systems

640

(1)

8. Why is it useful?

641

(8)

8.1. Gibbs formalism

641

(2)

8.2. An application: Jarzynski's identity

643

(5)

8.3. Response relations

648

(1)

9. What is missing-among other things?

649

(3)

9.1. The problem

649

(2)

9.2. The missing link

651

(1)

9.3. Example

652

(1)

References

652

(5)

Course 14. Mathematical aspects of the abelian sandpile model,

by Frank Redig

657

(74)

1. Introduction

661

(3)

2. Prelude: the one-dimensional model

664

(11)

2.1. The crazy office

664

(2)

2.2. Rooted spanning trees

666

(1)

2.3. Group structure

667

(5)

2.4. The stationary measure

672

(1)

2.5. Toppling numbers

673

(2)

2.6. Exercises

675

(1)

3. General finite volume abelian sandpiles

675

(18)

3.1. Allowed configurations

679

(3)

3.2. Rooted spanning trees

682

(2)

3.3. Group structure

684

(1)

3.4. Addition of recurrent configurations

685

(1)

3.5. General toppling matrices

686

(1)

3.6. Avalanches and waves

687

(2)

3.7. Height one and weakly allowed clusters

689

(4)

4. Towards infinite-volume: the basic questions

693

(9)

4.1. General estimates

696

(2)

4.2. Construction of a stationary Markov process

698

(4)

5. Infinite volume limits of the dissipative model

702

(8)

5.1. Ergodicity of the dissipative infinite-volume dynamics

706

(4)

6. Back to criticality

710

(10)

6.1. Wilson's algorithm and two-component spanning trees

713

(5)

6.2. A second proof of finiteness of waves

718

(2)

7. Stabilizability and "the organization of self-organized criticality"

720

(6)

8. Recommended literature

726

(1)

References

726

(5)

Course 15. Gibbsianness and non-Gibbsianness in lattice random fields,

by Roberto Fernández

731

(70)

1. Historical remarks and purpose of the course

735

(2)

2. Setup, notation, and basic notions

737

(5)

3. Probability kernels, conditional probabilities, and statistical mechanics

742

(10)

3.1. Probability kernels

742

(2)

3.2. Conditional probabilities

744

(3)

3.3. Specifications. Consistency

747

(5)

4. What it takes to be Gibbsian

752

(18)

4.1. Boltzmann prescription. Gibbs measures

752

(2)

4.2. Properties of Gibbsian (and some other) specifications

754

(4)

4.3. The Gibbsianness question

758

(9)

4.3.1. Construction of the vacuum potential

759

(3)

4.3.2. Summability of the vacuum potential

762

(2)

4.3.3. Kozlov theorem

764

(3)

4.4. Less Gibbsian measures

767

(3)

5. What it takes to be non-Gibbsian

770

(24)

5.1. Linear transformations of measures

770

(3)

5.2. Absence of uniform non-nullness

773

(1)

5.3. Absence of quasilocality

774

(3)

5.4. Surprise number one: renormalization maps

777

(10)

5.4.1. The scenarios

777

(2)

5.4.2. Step zero: understanding the conditioned measures

779

(3)

5.4.3. The three steps of a non-quasilocality proof

782

(4)

5.4.4. Non-quasilocality throughout the phase diagram

786

(1)

5.5. Surprise number two: spin-flip evolutions

787

(4)

5.6. Surprise number three: disordered models

791

(3)

References

794

(7)

Course 16. Simulation of statistical mechanics models,

by Alan D. Sokal

801

1. Overview

805

(1)

2. The Swendsen–Wang algorithm: some recent progress

805

(8)

2.1. Fortuin–Kasteleyn representation and Swendsen–Wang algorithm

806

(3)

2.2. Dynamic critical behavior: numerical results

809

(1)

2.3. How much do we understand about the SW dynamics?

810

(3)

References

813

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