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1 From Brownian motion to Euclidean fields 1

1.1 Brownian motion 1

1.1.1 Random walks 1

1.1.2 The sum over paths 9

1.1.3 The dimension two of Brownian curves 12

1.2 Euclidean fields 21

1.2.1 Free fields 22

1.2.2 Interacting fields and random walks 25

1.2.3 Self-avoiding walks and the limit n→0 31

1.2.4 Comparison with the high temperature expansion 33

1.2.5 The one-dimensional case 37

1.A Lattices 43

Notes 47

2 Grassmannian integrals and the two-dimensional Ising model 48

2.1 Grassmannian integrals 48

2.1.1 Anticommuting variables 48

2.1.2 Integrals 52

2.2 The two-dimensional Ising model 58

2.2.1 Duality 59

2.2.2 Transfer matrix 62

2.2.3 Fermionic representation 63

2.2.4 Free energy 67

2.2.5 Spontaneous magnetization 72

2.2.6 Correlation function in the high temperature phase 78

2.2.7 Surface tension 85

2.3 Critical continuous theory 91

2.3.1 Effective action 91

2.3.2 Correlation functions 95

2.A Quadratic differences and Painlevé equations 98

Notes 104

3 Spontaneous symmetry breaking,mean field 107

3.1 Mean field approximation 108

3.1.1 Dielectric constant of a polarizable medium 108

3.1.2 Classical spin model with a finite symmetry group 112

3.1.3 Continuous symmetry group 118

3.1.4 The Bethe approximation 121

3.1.5 Critical exponents 125

3.2 Lee-Yang zeroes 131

3.2.1 The Lee-Yang theorem 132

3.2.2 The one-dimensional case 135

3.2.3 General properties 136

3.2.4 Zeroes in the temperature plane 139

3.3 Large n limit 140

3.3.1 Saddle point method 141

3.3.2 Factorization 145

3.3.3 Coupling to an external field 148

3.4 Corrections to mean field 151

3.4.1 Laplace transform 153

Notes 159

4 Scaling transformations and the XY-model 162

4.1 Scaling laws.Real space renormalization 162

4.1.1 Homogeneity and scale invariance 162

4.1.2 Recurrence relations in real space 168

4.1.3 Examples and approximations 176

4.2 The XY-model 193

4.2.1 High temperature behaviour 195

4.2.2 Low temperature expansion.Vortices 197

4.2.3 The Villain action 204

4.2.4 Correlations 207

4.2.5 Renormalization flow 213

4.A Two-dimensional systems with continuous symmetry 219

4.A.1 Magnetization inequality 219

4.A.2 Correlation inequality 222

4.B Phenomenological renormalization 224

Notes 230

5 Continuous field theory and the renormalization group 233

5.1 The Lagrangian and dimensional analysis 233

5.1.1 Introduction 233

5.1.2 Generating functionals and dimensional analysis 236

5.2 The perturbative method 242

5.2.1 Diagrammatic series 242

5.2.2 Loop expansion 246

5.2.3 Evaluation of integrals and dimensional continuation 250

5.2.4 Group theoretical factors 259

5.2.5 Power counting 261

5.2.6 Perturbative renormalization 264

5.3 The renormalization group 270

5.3.1 Renormalization flow 270

5.3.2 Critical exponents 280

5.3.3 From the Gaussian ultraviolet fixed point to the infrared critical point in dimension less than four 282

5.3.4 Correlation functions at the critical point 284

5.3.5 Expansion near the critical point 290

5.3.6 Scaling laws below the critical temperature 296

5.4 Corrections to scaling laws 301

5.4.1 Deviation from the critical point in dimension lower than four 301

5.4.2 Logarithmic corrections in dimension four 303

5.4.3 Irrelevant operators 307

5.5 Numerical results 310

5.5.1 ε-expansion of critical exponents 311

5.5.2 Equation of state 313

5.5.3 Amplitude ratios 314

5.5.4 Three-dimensional results 317

5.A Multicritical points 317

Notes 326

6 Lattice gauge fields 328

6.1 Generalities 328

6.1.1 Presentation 328

6.1.2 The continuous limit 332

6.1.3 Order parameter and Elitzur's theorem 341

6.1.4 Duality 345

6.2 Structure of the phase diagram 352

6.2.1 Mean field approximation 352

6.2.2 Corrections to mean field and restoration of gauge invariance 359

6.2.3 Discrete groups:1/d expansion 362

6.2.4 Continuous groups:computation of corrections 365

6.3 Strong coupling expansions 371

6.3.1 Convergence 371

6.3.2 Character expansions 374

6.3.3 Free energy 380

6.3.4 String tension and roughening transition 385

6.3.5 Mass spectrum 390

6.4 Lattice fermions 393

6.4.1 The doubling problem 393

6.4.2 The Nielsen-Ninomiya theorem 396

6.4.3 Staggered fermions 399

Notes 401

Index 405

7 Diagrammatic methods 405

8 Numerical simulations 456

9 Conformal invariance 501

10 Disordered systems and fermionic methods 646

11 Random geometry 738