Mathematical Statistical Physics (eBook)

Front cover 1
Lecturers who contributed to this volume 3
Title page 4
Copyright page 5
Previous sessions 7
Organizers 10
Lecturers 10
Participants 12
Preface 16
Informal seminars 20
Table of contents 22
Course 1 Random matrices and determinantal processes 34
Introduction 38
Point processes 38
General theory 38
Determinantal processes 46
Measures defined by products of several determinants 52
Non-intersecting paths and the Aztec diamond 56
Non-intersecting paths and the LGV theorem 56
The Aztec diamond 58
Relations to other models 63
Asymptotics 66
Double contour integral formula for the correlation kernel 66
Asymptotics for the Aztec diamond 68
Asymptotics for random permutations 74
The corner growth model 76
Mapping to non-intersecting paths 76
The Schur and Plancherel measures 78
A discrete polynuclear growth model 81
Proof of theorem 5.1 83
References 86
Course 2 Some recent aspects of random conformally invariant systems 90
Some discrete models 96
Self-avoiding walks and polygons 96
Random walk loops 97
Site-percolation 97
The Ising model 99
The Potts models 101
FK representations of Potts models 101
The O(N) models 102
Conformal invariance 104
A "conformal Haar measure" on self-avoiding loops 106
Preliminaries 106
A conformal invariance property 106
Uniqueness 107
Existence 110
Schramm-Loewner Evolutions 113
Definition 113
Computing with SLE 116
Conformal loop-ensembles 120
Definition 120
First properties 122
The loop-soup construction 123
The Gaussian free field 127
Definition 127
"Cliffs" as level lines 128
References 130
Course 3 Conformal random geometry 134
Preamble 138
Introduction 140
A brief conformal history 140
Conformal geometrical structures 143
Quantum gravity 144
Stochastic Löwner evolution 145
Recent developments 146
Synopsis 148
Intersections of random walks 150
Non-intersection probabilities 150
Quantum gravity 153
Random walks on a random lattice 156
Non-intersections of packets of walks 165
Mixing random & self-avoiding walks
General star configurations 170
Quantum gravity for SAW's & RW's
RW-SAW exponents 178
Brownian hiding exponents 179
Percolation clusters 181
Cluster hull and external perimeter 181
Harmonic measure of percolation frontiers 183
Harmonic and path crossing exponents 184
Quantum gravity for percolation 185
Multifractality of percolation clusters 186
Conformally invariant frontiers and quantum gravity 189
Harmonic measure and potential near a fractal frontier 189
Calculation of multifractal exponents from quantum gravity 193
Geometrical analysis of multifractal spectra 200
Higher multifractal spectra 206
Double-sided spectra 206
Higher multifractality of multiple path vertices 211
Winding of conformally invariant curves 211
Harmonic measure and rotations 212
Exact mixed multifractal spectra 213
Conformal invariance and quantum gravity 216
Rotation scaling exponents 218
Legendre transform 220
O(N) & Potts models and the Stochastic Löwner Evolution
Geometric duality in O(N) and Potts cluster frontiers 220
Geometric duality of SLEkappa 223
Quantum gravity duality and SLE 226
Dual dimensions 226
KPZ for SLE 229
Short distance expansion 231
Multiple paths in O(N), Potts models and SLE 233
SLE(kappa, rho) and quantum gravity 235
Multifractal exponents for multiple SLE's 237
References 242
Course 4 Random motions in random media 252
Introduction 256
RWRE 257
RCM 258
Back to RWRE 262
Diffusions in random environment 268
References 270
Course 5 An introduction to mean field spin glas theory: methods and results 276
Introduction 280
The mean field ferromagnetic model. Convexity and cavity methods 284
The mean field spin glass model. Basic definitions 288
The interpolation method and its generalizations 290
The thermodynamic limit and the variational bounds 293
The Parisi representation for the free energy 296
Conclusion and outlook for future developments 301
References 302
Course 6 Short-range spin glasses: selected open problems 306
Introduction 310
The Fortuin-Kasteleyn random cluster representation and phase transitions 314
Spin glass ground states and invasion percolation 317
Ground state multiplicity in the 2D EA spin glass 321
References 323
Course 7 Computing the number of metastable states in infinite-range models 328
Introduction 332
The TAP equations 336
A simple analysis of the solutions of the Bethe equations 340
The direct approach: general considerations 345
The supersymmetric formulation 349
Spontaneous supersymmetry breaking 351
An explicit computation: the complexity of the SK model 354
A few words on quenched complexity 357
Conclusions and open problems 358
References 360
Course 8 Dynamics of trap models 364
Introduction 368
Definition of the Bouchaud trap model 371
Examples of trap models 374
Natural questions on trap models 375
References 376
The one-dimensional trap model 377
The Fontes-Isopi-Newman singular diffusion 377
The scaling limit 379
Time-scale change of Brownian motion 380
Convergence of the fixed-time distributions 382
A coupling for walks on different scales 382
Scaling limit 384
Aging results 385
Subaging results 386
Behaviour of the aging functions on different time scales 387
References 389
The trap model in dimension larger than one 389
The fractional-kinetics process 389
Scaling limit 391
Aging results 392
The coarse-graining procedure 393
References 396
The arcsine law as a universal aging scheme 396
Aging on large complete graphs 397
Deep traps 400
Shallow traps 402
Very deep traps 404
Proof of Theorem 5.1 404
The alpha-stable subordinator as a universal clock 406
Potential-theoretic characterisation 410
Applications of the arcsine law 413
Aging in the REM 413
Short time scales 414
Long time scales 416
Open questions and conjectures 418
Aging on large tori 419
Appendix A. Subordinators 420
References 424
Course 9 Quantum entropy and quantum information 428
Introduction 432
Rudiments of Classical Information Theory 433
Entropy in Classical Information Theory 438
Entropy of a pair of random variables 439
Shannon's Noiseless Channel Coding Theorem 442
Asymptotic Equipartition Property (AEP) 443
Consequences of the AEP 445
Information transmission and Channel Capacity 447
Introduction to Quantum Information Theory 450
Open systems 453
Properties of the density matrix 454
Reduced density matrix and partial trace 455
Time evolution of open systems 460
Generalized measurements 463
Implementation of a generalized measurement 464
Quantum entropy 467
Properties of the von Neumann entropy S(rho) 469
Data compression in Quantum Information Theory 474
Schumacher's Theorem for memoryless quantum sources 475
Quantum channels and additivity 481
The noise in the channel 483
Capacities of a quantum channel 484
Classical capacity of a quantum channel 485
A sufficient condition for additivity 491
Multiplicativity of the maximal p-norm 495
Bibliography 496
Course 10 Two lectures on iterative coding and statistical mechanics 500
Introduction 504
Codes on graphs 505
A simple-minded bound and belief propagation 506
Characterizing the code performances 506
Bounding the conditional entropy 507
A parenthesis 511
Density evolution a.k.a. distributional recursive equations 512
The area theorem and some general questions 515
Historical and bibliographical note 517
References 517
Course 11 Evolution in fluctuating populations 522
Introduction 526
Some classical coalescent theory 528
Kingman’s coalescent 528
Variable population size 530
Introducing structure 530
Fluctuations matter 531
Balancing selection 532
A second neutral locus 535
The problem 536
The ancestral recombination graph and local trees 537
Back to the main plot 539
The diffusion approximation 540
Extensions 545
Summary and notes of caution 546
Spatial structure and the Malécot formula 547
Branching process models 547
The classical stepping stone model 550
Duality 552
Neutral evolution 554
Random walks in random environments 556
Models in continuous space 556
Malécot’s formula 557
Spatial models 562
Locally regulated populations 562
Competing species 566
Branching annihilating random walk 570
A duality and a conjecture for Model II 571
Conjectures for Model I 572
Heteromyopia 573
References 575
Course 12 Multi-scale analysis of population models 580
Spatial diffusion models of population genetics 584
Duality and representation via coalescent processes 589
The function-valued dual process 589
The refined dual process 593
Application: ergodic theorem 598
Historical processes 599
The neutral models: IMM and IFWD 599
The concept of the historical process 601
The concept of the look-down process 602
Representation theorem via look-down 605
Representation via coalescent 608
Quasi-equilibria: elements of the multi-scale analysis 610
Mean-field models and McKean-Vlasov limit 611
The mean-field dual 612
Longtime behavior of McKean-Vlasov process 614
Large time-space scale behavior of mean-field model 615
Punctuated equilibria and hierarchical mean-field limit 618
Detailed specification of selection and mutation mechanisms 619
Interpretation of the model from the point of view of evolutionary theory 621
The basic scenario 622
The hierarchical mean-field limit 624
Two interaction chains 626
Basic limit theorem 627
Punctuated equilibria: emergence and take-over 629
References 635
Course 13 Elements of nonequilibrium statistical mechanics 640
Elements of introduction 644
The goal 644
The plan 645
Elements of an H-theorem 646
Heuristics of an H-theorem 646
H-theorem: the problem 647
Macroscopic H-theorem 648
Semigroup property 649
Autonomy on a finite partition 650
Reversibility 650
Propagation of constrained equilibrium 651
Pathwise H-theorem 652
Elements of heat conduction 654
Hamiltonian dynamics 654
Non-uniqueness 655
Boundary conditions to the Hamiltonian flow 656
Heat bath dynamics 657
Entropy production and heat currents 657
Mean entropy production 659
Strict positivity of mean entropy production 660
Getting work done 660
Lagrangian approach 662
First goal 662
Entropy in irreversible thermodynamics 663
A little entropology 665
Reduced variables 665
Distributions 666
Entropy functional 666
Microcanonical entropy 667
Equilibrium entropy 667
Dynamical entropy 668
Closed systems 668
Extensions 669
Example: diffusion in local thermodynamic equilibrium 669
Open systems 671
Heat conduction 671
Asymmetric exclusion process 671
Strongly chaotic dynamical systems 673
Why is it useful? 674
Gibbs formalism 674
An application: Jarzynski's identity 676
Response relations 681
What is missing - among other things? 682
The problem 682
The missing link 684
Example 685
References 685
Course 14 Mathematical aspects of the abelian sandpile model 690
Introduction 694
Prelude: the one-dimensional model 697
The crazy office 697
Rooted spanning trees 699
Group structure 700
The stationary measure 705
Toppling numbers 706
Exercises 708
General finite volume abelian sandpiles 708
Allowed configurations 712
Rooted spanning trees 715
Group structure 717
Addition of recurrent configurations 718
General toppling matrices 719
Avalanches and waves 720
Height one and weakly allowed clusters 722
Towards infinite-volume: the basic questions 726
General estimates 729
Construction of a stationary Markov process 731
Infinite volume limits of the dissipative model 735
Ergodicity of the dissipative infinite-volume dynamics 739
Back to criticality 743
Wilson's algorithm and two-component spanning trees 746
A second proof of finiteness of waves 751
Stabilizability and ``the organization of self-organized criticality'' 753
Recommended literature 759
References 759
Course 15 Gibbsianness and non-Gibbsianness in lattice random fields 764
Historical remarks and purpose of the course 768
Setup, notation, and basic notions 770
Probability kernels, conditional probabilities, and statistical mechanics 775
Probability kernels 775
Conditional probabilities 777
Specifications. Consistency 780
What it takes to be Gibbsian 785
Boltzmann prescription. Gibbs measures 785
Properties of Gibbsian (and some other) specifications 787
The Gibbsianness question 791
Construction of the vacuum potential 792
Summability of the vacuum potential 795
Kozlov theorem 797
Less Gibbsian measures 800
What it takes to be non-Gibbsian 803
Linear transformations of measures 803
Absence of uniform non-nullness 806
Absence of quasilocality 807
Surprise number one: renormalization maps 810
The scenarios 810
Step zero: understanding the conditioned measures 812
The three steps of a non-quasilocality proof 815
Non-quasilocality throughout the phase diagram 819
Surprise number two: spin-flip evolutions 820
Surprise number three: disordered models 824
References 827
Course 16 Simulation of statistical mechanics models 834
Overview 838
The Swendsen-Wang algorithm: some recent progress 838
Fortuin-Kasteleyn representation and Swendsen-Wang algorithm 839
Dynamic critical behavior: numerical results 842
How much do we understand about the SW dynamics? 843
References 846