Lectures on Noise Sensitivity and Percolation

Overview......Page 7
2 Some Examples......Page 11
3 Pivotality and Influence......Page 13
4 The Kahn, Kalai, Linial Theorem......Page 14
6 Benjamini, Kalai and Schramm noise sensitivity Theorem......Page 16
7 Percolation crossings: our final and most important example......Page 18
1 The model......Page 23
2 Russo-Seymour-Welsh......Page 24
4 Conformal invariance at criticality and SLE processes......Page 25
5 Critical exponents......Page 27
6 Quasi-multiplicativity......Page 28
1 Monotone functions and the Margulis-Russo formula......Page 29
3 Sharp thresholds in general : the Friedgut-Kalai Theorem......Page 30
4 The critical point for percolation for Z2 and T is 12......Page 31
5 Further discussion......Page 32
1 Discrete Fourier analysis and the energy spectrum......Page 35
2 Examples......Page 36
3 Noise sensitivity and stability in terms of the energy spectrum......Page 37
4 Link between the spectrum and influence......Page 38
5 Monotone functions and their spectrum......Page 39
1 Tensorization......Page 43
2 The one-dimensional case (n=1)......Page 44
3 Proof of the KKL Theorems......Page 46
4 KKL away from the uniform measure......Page 49
5 The noise sensitivity theorem......Page 51
Appendix on Bonami-Gross-Beckner......Page 53
1 Influences of crossing events......Page 59
2 The case of Z2 percolation......Page 63
3 Some other consequences of our study of influences......Page 66
4 Quantitative noise sensitivity......Page 68
1 The model of first passage percolation......Page 75
3 The case of the torus......Page 77
5 Further discussion......Page 80
2 The Revealment Theorem......Page 85
3 An application to noise sensitivity of percolation......Page 89
4 Lower bounds on revealments......Page 91
5 An application to a critical exponent......Page 93
6 Does noise sensitivity imply low revealment?......Page 94
1 Definition of the spectral sample......Page 99
2 A way to sample the spectral sample in a sub-domain......Page 101
3 Nontrivial spectrum near the upper bound for percolation......Page 103
1 State of the art and main statement......Page 109
2 Overall strategy......Page 111
3 Toy model: the case of fractal percolation......Page 113
4 Back to the spectrum: an exposition of the proof......Page 120
5 The radial case......Page 130
1 The model of dynamical percolation......Page 135
2 What's going on in high dimensions: Zd, d19?......Page 136
4 The second moment method and the spectrum......Page 137
5 Proof of existence of exceptional times on T......Page 139
6 Exceptional times via the geometric approach......Page 142