PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

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230 <strong>Geoffrey</strong> Grimmett for crossing probabilities between two sub-intervals of a simple closed curve C. Let C be a simple closed curve, and let z1, z2, z3, z4 be four points on C in clockwise order. There is a conformal map φ on the interior of C which maps to the unit disc, taking C to its circumference, and the points zi to the points wi. There are many such maps, but the cross-ratio of such maps, (9.13) u = (w4 − w3)(w2 − w1) (w3 − w1)(w4 − w2) , is a constant satisfying 0 ≤ u ≤ 1 (we think of zi and wi as points in the complex plane). We may parametrise the wi as follows: we may assume that w1 = e iθ , w2 = e −iθ , w3 = −e iθ , w4 = −e −iθ for some θ satisfying 0 ≤ θ ≤ π 2 . Note that u = sin2 θ. We take α to be the segment of C from z1 to z2, and β the segment from z3 to z4. Using the hypothesis of conformal invariance, we have that π(G) = π(φG), where G = (C; α, β; ∅, ∅), implying that π(G) may be expressed as some function f(u), where u is given in (9.13). Cardy has derived a differential equation for f, namely (9.14) u(1 − u)f ′′ (u) + 2 3 (1 − 2u)f ′ (u) = 0, together with the boundary conditions f(0) = 0, f(1) = 1. The solution is a hypergeometric function, (9.15) f(u) = 3Γ(2 3 ) Γ( 1 3 )2 u1/3 2F1( 1 2 3 , 3 4 , 3 ; u). Recall that u = sin 2 θ. The derivation is somewhat speculative, but the predictions of the formula may be verified by Monte Carlo simulation (see Figure 3.2 of [232]). The above ‘calculation’ is striking. Similar calculations may well be possible for more complicated crossing probabilities than the case treated above. See, for example, [10, 344]. In the above formulation, the principle of conformal invariance is expressed in terms of a collection {π(G)} of limiting ‘crossing probabilities’. It would be useful to have a representation of these π(G) as probabilities associated with a specific random variable on a specific probability space. Aizenman [10] has made certain proposals about how this might be possible. In his formulation, we observe a bounded region DR = [0, R] 2 , and we shrink the lattice spacing a of bond percolation restricted to this domain. Let p = pc, and let Ga be the graph of open connections of bond percolation with lattice spacing a on DR. By describing Ga through the set of Jordan curves describing the realised paths, he has apparently obtained sufficient compactness to imply
Percolation and Disordered Systems 231 the existence of weak limits as a → 0. Possibly there is a unique weak limit, and Aizenman has termed an object sampled according to this limit as the ‘web’. The fundamental conjectures are therefore that there is a unique weak limit, and that this limit is conformally invariant. The quantities π(G) should then arise as crossing probabilities in ‘webmeasure’. This geometrical vision may be useful to physicists and mathematicians in understanding conformal invariance. In one interesting ‘continuum’ percolation model, conformal invariance may actually be proved rigorously. Drop points {X1, X2, . . . } in the plane R2 in the manner of a Poisson process with intensity λ. Now divide R2 into tiles {T(X1), T(X2), . . . }, where T(X) is defined as the set of points in R2 which are no further from X than they are from any other point of the Poisson process (this is the ‘Voronoi tesselation’). We designate each tile to be open with probability 1 2 and closed otherwise. This continuum percolation model has a property of self-duality, and it inherits properties of conformal invariance from those of the underlying Poisson point process. See [10, 56].
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PERCOLATION AND DISORDERED SYSTEMS
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144 Geoffrey Grimmett ÈÊÇ��
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146 Geoffrey Grimmett 8. Critical P
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148 Geoffrey Grimmett θ(p) 1 pc 1
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150 Geoffrey Grimmett physical imag
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152 Geoffrey Grimmett Fig. 2.1. Par
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154 Geoffrey Grimmett Fig. 2.2. An
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156 Geoffrey Grimmett 3.1 Percolati
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158 Geoffrey Grimmett Kesten [201]
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160 Geoffrey Grimmett whence d dp P
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162 Geoffrey Grimmett Fig. 4.1. An
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164 Geoffrey Grimmett s 1 θ = 0 pc
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166 Geoffrey Grimmett ∂B(n) 0 e f
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168 Geoffrey Grimmett Fig. 4.5. A s
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170 Geoffrey Grimmett Theorem 5.5 (
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172 Geoffrey Grimmett Proof of Theo
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174 Geoffrey Grimmett 5.3 Site and
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176 Geoffrey Grimmett 6.1 Using Sub
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178 Geoffrey Grimmett Now φ(p) = 0
Page 40 and 41: 180 Geoffrey Grimmett so that (3.10
Page 42 and 43: 182 Geoffrey Grimmett x 2 y 2 x 4 x
Page 44 and 45: 184 Geoffrey Grimmett Similarly, Pp
Page 46 and 47: 186 Geoffrey Grimmett It remains to
Page 48 and 49: 188 Geoffrey Grimmett Fix β < pc a
Page 50 and 51: 190 Geoffrey Grimmett where δ 2 =
Page 52 and 53: 192 Geoffrey Grimmett Fig. 7.1. Tak
Page 54 and 55: 194 Geoffrey Grimmett 7.2 Percolati
Page 56 and 57: 196 Geoffrey Grimmett 4. The open c
Page 58 and 59: 198 Geoffrey Grimmett For n > m, le
Page 60 and 61: 200 Geoffrey Grimmett Lemma 7.17. I
Page 62 and 63: 202 Geoffrey Grimmett Lemma 7.24. S
Page 64 and 65: 204 Geoffrey Grimmett 000 111 000 1
Page 66 and 67: 206 Geoffrey Grimmett Fig. 7.7. An
Page 68 and 69: 208 Geoffrey Grimmett Fig. 7.9. The
Page 70 and 71: 210 Geoffrey Grimmett Fig. 7.10. Il
Page 72 and 73: 212 Geoffrey Grimmett 8.1 Percolati
Page 74 and 75: 214 Geoffrey Grimmett may be shown
Page 76 and 77: 216 Geoffrey Grimmett Lemma 8.11. L
Page 78 and 79: 218 Geoffrey Grimmett 0 u noting th
Page 80 and 81: 220 Geoffrey Grimmett where α ′
Page 82 and 83: 222 Geoffrey Grimmett The infra-red
Page 84 and 85: 224 Geoffrey Grimmett 9. PERCOLATIO
Page 86 and 87: 226 Geoffrey Grimmett Fig. 9.2. If
Page 88 and 89: 228 Geoffrey Grimmett x Fig. 9.3. I
Page 92 and 93: 232 Geoffrey Grimmett 10. RANDOM WA
Page 94 and 95: 234 Geoffrey Grimmett the volume of
Page 96 and 97: 236 Geoffrey Grimmett We define sim
Page 98 and 99: 238 Geoffrey Grimmett We now use th
Page 100 and 101: 240 Geoffrey Grimmett p+ 1 pc(site)
Page 102 and 103: 242 Geoffrey Grimmett Fig. 10.6. Th
Page 106 and 107: 246 Geoffrey Grimmett A L d -path i
Page 108 and 109: 248 Geoffrey Grimmett where pc(site
Page 110 and 111: 250 Geoffrey Grimmett now make two
Page 116 and 117: 256 Geoffrey Grimmett II. P(C �=
Page 118 and 119: 258 Geoffrey Grimmett Theorem 11.13
Page 120 and 121: 260 Geoffrey Grimmett (b) Under wha
Page 122 and 123: 262 Geoffrey Grimmett is obtained b
Page 124 and 125: 264 Geoffrey Grimmett 13.2 Weak Lim
Page 126 and 127: 266 Geoffrey Grimmett where p = 1
Page 128 and 129: 268 Geoffrey Grimmett Evidently pg(
Page 130 and 131: 270 Geoffrey Grimmett That is to sa
Page 132 and 133: 272 Geoffrey Grimmett which, via Le
Page 134 and 135: 274 Geoffrey Grimmett Turning to th
Page 136 and 137: 276 Geoffrey Grimmett Each circuit
Page 138 and 139: 278 Geoffrey Grimmett If A(q) < 1 (
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280 Geoffrey Grimmett REFERENCES 1.
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282 Geoffrey Grimmett 30. Alexander
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284 Geoffrey Grimmett 63. Berg, J.
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286 Geoffrey Grimmett 93. Chayes, J
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288 Geoffrey Grimmett 123. Durrett,
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290 Geoffrey Grimmett 158. Grimmett
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292 Geoffrey Grimmett 191. Higuchi,
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294 Geoffrey Grimmett 224. Koteck´
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296 Geoffrey Grimmett 259. Meester,
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298 Geoffrey Grimmett 290. Newman,
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300 Geoffrey Grimmett 323. Roy, R.
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302 Geoffrey Grimmett 355. Wierman,
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PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

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