Lectures on Probability Theory and Statistics: Ecole d’Ete de Probabilites de Saint-Flour XXXII - 2002

Cover......Page 1
Series: Lecture Notes in Mathematics 1840......Page 2
Lectures on Probability Theory and Statistics: Ecole d’Eté de Probabilités de Saint-Flour XXXII - 2002......Page 4
Copyright - ISBN: 9783540213161......Page 5
Preface......Page 6
Contents......Page 8
Part I - Boris Tsirelson: Scaling Limit, Noise, Stability......Page 10
Contents......Page 12
Introduction......Page 14
1.1 Two Toy Models......Page 15
1.2 Our Limiting Procedures......Page 17
1.3 Examples of High Symmetry......Page 20
1.4 Example of Low Symmetry......Page 21
1.5 Trees, Not Cubes......Page 23
1.6 Sub-$ igma$-fields......Page 24
2.1 More on Our Limiting Procedures......Page 26
2.2 Coarse Probability Space: Definition and Simple Example......Page 29
2.3 Good Use of Joint Compactification......Page 31
3.1 Product of Coarse Probability Spaces......Page 37
3.2 Dyadic Case......Page 39
3.3 Scaling Limit of Fourier-Walsh Coefficients......Page 43
3.4 The Limiting Object......Page 46
3.5 Time Shift; Noise......Page 50
4.1 Three Discrete Semigroups: Algebraic Definition......Page 53
4.3 Random Walks and Stochastic Flows in Discrete Semigroups......Page 54
4.4 Three Continuous Semigroups......Page 57
4.5 Convolution Semigroups in These Continuous Semigroups......Page 58
4.6 Getting Dyadic......Page 59
4.7 Scaling Limit......Page 60
4.8 Noises......Page 61
4.9 The Poisson Snake......Page 62
5.1 Discrete Case......Page 65
5.2 Continuous Case......Page 68
5.3 Back to Discrete: Two Kinds of Stability......Page 74
6.1 First Chaos, Decomposable Processes, Stability......Page 76
6.2 Higher Levels of Chaos......Page 79
6.3 An Old Question of Jacob Feldman......Page 84
6.4 Black Noise......Page 87
7.1 Convolution Semigroup of the Brownian Web......Page 92
7.2 Some General Arguments......Page 93
7.3 The Key Argument......Page 94
7.4 Remarks......Page 98
7.5 A Combinatorial By-product......Page 99
8.1 Beyond the One-Dimensional Time......Page 102
8.2 The ‘Wave Noise’ Approach......Page 103
8.3 Groups, Semigroups, Kernels......Page 105
8.4 Abstract Nonsense of Le Jan-Raimond’s Theory......Page 107
References......Page 112
Index......Page 114
Part II - Wendelin Werner: Random Planar Curves and Schramm-Loewner Evolutions......Page 116
Contents......Page 118
Foreword and Summary......Page 120
1.1 General Motivation......Page 122
1.2 Loop-Erased Random Walks......Page 124
1.3 Iterations of Conformal Maps and SLE......Page 126
1.4 The Critical Percolation Exploration Process......Page 128
1.5 Chordal versus Radial......Page 130
1.6 Conclusion......Page 131
2.1 Measuring the Size of Subsets of the Half-Plane......Page 133
2.2 Loewner Chains......Page 135
3.1 Definition......Page 139
3.2 A First Computation......Page 140
3.3 Chordal $SLE_{\kappa}$ in Other Domains......Page 142
3.4 Transience......Page 144
4.1 Image of SLE under Conformal Maps......Page 146
4.2 Locality for $SLE_6$......Page 147
4.3 Restriction for $SLE_{8/3}$......Page 148
5.1 A Reflected Brownian Motion......Page 152
5.2 Brownian Excursions and $SLE_{8/3}$......Page 156
6.1 Definitions......Page 160
6.2 Relation between Radial and Chordal SLE......Page 161
6.3 Radial $SLE_6$ and Re.ected Brownian Motion......Page 163
7.1 Disconnection Exponents......Page 165
7.2 Derivative Exponents......Page 166
7.3 First Consequences......Page 168
8.2 Brownian Crossings......Page 170
8.3 Disconnection Exponent......Page 172
8.4 Other Exponents......Page 174
8.5 Hausdorff Dimensions......Page 175
9.2 Uniform Spanning Trees, Wilson’s Algorithm......Page 178
9.3 Convergence to Chordal $SLE_8$......Page 181
9.4 The Loop-Erased Random Walk......Page 183
10.1 Introduction......Page 185
10.2 The Cardy-Smirnov Formula......Page 187
10.3 Convergence to $SLE_6$ and Consequences......Page 190
11.1 A List of Ideas......Page 194
11.2 A List of Open Problems......Page 195
References......Page 199
List of Participants......Page 206
List of Short Lectures......Page 208