- Page 1: RANDOM FIELDS AND THEIR GEOMETRY Ro
- Page 4 and 5: iv Contents 2.5 Orthogonal expansio
- Page 6 and 7: 1 Random fields 1.1 Random fields a
- Page 8 and 9: 1.2 Gaussian fields 3 • In applic
- Page 12 and 13: 1.3 The Brownian family of processe
- Page 14 and 15: 1.3 The Brownian family of processe
- Page 16 and 17: 1.4 Stationarity 11 Then Aω and B
- Page 18 and 19: 1.4 Stationarity 13 operation +, if
- Page 20 and 21: 1.4 Stationarity 15 L 2 (P) ≡ L 2
- Page 22 and 23: 1.4 Stationarity 17 additional defi
- Page 24 and 25: s ∈ R N , set θ(g, t)(s) = As +
- Page 26 and 27: 1.4 Stationarity 21 and the terms i
- Page 28 and 29: W1 and W2, such that 26 (1.4.24) ft
- Page 30 and 31: moments, when they exist, are zero;
- Page 32 and 33: 1.4.5 Constant variance 1.4 Station
- Page 34 and 35: 1.4 Stationarity 29 ∆ where δij
- Page 36 and 37: Now use the spectral decomposition
- Page 38 and 39: 1.4 Stationarity 33 If T = R N unde
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- Page 44 and 45: 2 Gaussian fields The aim of this C
- Page 46 and 47: 2.1 Boundedness and continuity 41 b
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- Page 50 and 51: should be taken as the definition o
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- Page 54 and 55: 2.2 Examples 2.2.1 Fields on R N 2.
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2.2 Examples 55 The proof requires
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canonical metric d, where (2.2.15)
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2.2 Examples 59 Similarly, again re
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of a measure µ on RN , then by ana
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2.2 Examples 63 These inequalities
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2.2 Examples 65 Fix ε > 0 and supp
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2.3 Borell-TIS inequality 2.3 Borel
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2.3 Borell-TIS inequality 69 But th
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2.3 Borell-TIS inequality 71 the se
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2.3 Borell-TIS inequality 73 so tha
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2.4 Comparison inequalities 75 g, a
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2.5 Orthogonal expansions 77 which
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versa. Start with � S = u : T →
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2.5 Orthogonal expansions 81 With S
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2.5 Orthogonal expansions 83 where
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2.5 Orthogonal expansions 85 where
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2.6 Majorising measures 87 is alway
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2.6 Majorising measures 89 replacin
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2.6 Majorising measures 91 This Lem
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Finally, set Λ = min(Dµ, Dν). Th
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3 Geometry For this Chapter we are
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3.2 Basic integral geometry 97 fiel
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3.2 Basic integral geometry 99 Now
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and define (3.2.9) ϕ(A) = � h(A,
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3.3 Excursion sets again 103 drille
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espect to T at the level u. (3.3.4)
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3.3 Excursion sets again 107 this p
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3.3 Excursion sets again 109 will c
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3.3 Excursion sets again 111 FIGURE
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3.4 Intrinsic volumes 113 suitable
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3.4 Intrinsic volumes 115 Comparing
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functional ˜ ψu for which � �
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3.5 Manifolds and tensors 119 If a
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3.5 Manifolds and tensors 121 for a
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3.5 Manifolds and tensors 123 up th
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y (3.5.8) α ∧ β = (r + s)! A(α
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3.5 Manifolds and tensors 127 and t
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3.6 Riemannian manifolds 129 Simila
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3.6 Riemannian manifolds 131 The ba
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3.6 Riemannian manifolds 133 If θ
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3.6 Riemannian manifolds 135 Given
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3.6 Riemannian manifolds 137 manifo
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3.6 Riemannian manifolds 139 FIGURE
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3.6 Riemannian manifolds 141 and we
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3.6 Riemannian manifolds 143 With t
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3.6 Riemannian manifolds 145 We are
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3.7 Piecewise smooth manifolds 147
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3.7 Piecewise smooth manifolds 149
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3.7 Piecewise smooth manifolds 151
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3.7 Piecewise smooth manifolds 153
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and the homeomorphisms ψt are such
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3.8 Intrinsic volumes again 157 We
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3.8 Intrinsic volumes again 159 to
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3.9 Critical Point Theory 161 C 2 f
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� k 3.9 Critical Point Theory 163
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3.9 Critical Point Theory 165 What
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3.9 Critical Point Theory 167 [0, 1
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3.9 Critical Point Theory 169 and t
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4 Gaussian random geometry With the
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4.1 An expectation meta-theorem 173
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4.1 An expectation meta-theorem 175
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have 4.1 An expectation meta-theore
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4.1 An expectation meta-theorem 179
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4.1 An expectation meta-theorem 181
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4.1 An expectation meta-theorem 183
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4.1 An expectation meta-theorem 185
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4.2 Suitable regularity and Morse f
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our manifold as usual as (4.2.2) 4.
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4.4 Higher moments 191 Take α(t) =
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4.5 Preliminary Gaussian computatio
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implying 4.5 Preliminary Gaussian c
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4.6 Mean Euler characteristics: Euc
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4.6 Mean Euler characteristics: Euc
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all out again. We start with 4.6 Me
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4.6 Mean Euler characteristics: Euc
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4.6 Mean Euler characteristics: Euc
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4.6 Mean Euler characteristics: Euc
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4.7 The meta-theorem on manifolds 2
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4.7 The meta-theorem on manifolds 2
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4.8 Riemannian structure induced by
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4.8 Riemannian structure induced by
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4.8 Riemannian structure induced by
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4.8 Riemannian structure induced by
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4.9 Another Gaussian computation 22
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4.10.1 Manifolds without boundary 4
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4.10 Mean Euler characteristics: Ma
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4.10 Mean Euler characteristics: Ma
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4.10 Mean Euler characteristics: Ma
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4.11 Examples 231 Stationary fields
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4.11 Examples 233 its own curvature
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4.12 Chern-Gauss-Bonnet Theorem 4.1
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References [1] R. J. Adler. The Geo
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References 283 [25] R. M. Dudley. L
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References 285 [58] John M. Lee. In
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References 287 [88] D. Slepian. On