PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

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222 <strong>Geoffrey</strong> Grimmett The infra-red bound emerges in the proof of (c), of which there follows an extremely brief account. We write x ⇔ y, and say that x is ‘doubly connected’ to y, if there exist two edge-disjoint open paths from x to y. We express τp(0, x) in terms of the ‘doubly connected’ probabilities δp(u, v) = Pp(u ⇔ v). In doing so, we encounter formulae involving convolutions, which may be treated by taking transforms. At the first stage, we have that � � {0 ↔ x} = {0 ⇔ x} ∪ 〈u,v〉 � 0 ⇔ (u, v) ↔ x � � where � 0 ⇔ (u, v) ↔ x � represents the event that 〈u, v〉 is the ‘first pivotal edge’ for the event {0 ↔ x}, and that 0 is doubly connected to u. (Similar but more complicated events appear throughout the proof.) Therefore (8.27) τp(0, x) = δp(0, x) + � 〈u,v〉 Now, with A(0, u; v, x) = {v ↔ x off C 〈u,v〉(0)}, � � Pp 0 ⇔ (u, v) ↔ x . � � Pp 0 ⇔ (u, v) ↔ x � � = pPp 0 ⇔ u, A(0, u; v, x) � � � = pδp(0, u)τp(v, x) − pEp 1 {0⇔u} τp(v, x) − 1A(0,u;v,x) � whence, by (8.27), (8.28) τp(0, x) = δp(0, x) + δp ⋆ (pI) ⋆ τp(x) − Rp,0(0, x) where ⋆ denotes convolution, I is the nearest-neighbour function I(u, v) = 1 if and only if u ∼ v, and Rp,0 is a remainder. Equation (8.28) is the first step of the lace expansion, In the second step, the remainder Rp,0 is expanded similarly, and so on. Such further expansions yield the lace expansion: if p < pc then (8.29) τp(0, x) = hp,N(0, x) + hp,N ⋆ (pI) ⋆ τp(x) + (−1) N+1 Rp,N(0, x) for appropriate remainders Rp,N, and where hp,N(0, x) = δp(0, x) + N� (−1) j Πp,j(0, x) and the Πp,n are appropriate functions (see Theorem 4.2 of [180]) involving nested expectations of quantities related to ‘double connections’. j=1
Percolation and Disordered Systems 223 We take Fourier transforms of (8.29), and solve to obtain that (8.30) �τp = �δp + �N j=1 (−1)jΠp,j � + (−1) N+1Rp,N � 1 − p� I� δp − p� I �N j=1 (−1)j � . Πp,j The convergence of the lace expansion, and the consequent validity of this formula for �τp, is obtained roughly as follows. First, one uses the BK inequality to derive bounds for the δp, Πp,j, Rp,j in terms of the functions T(p) and W(p). These bounds then imply bounds for the corresponding transforms. In this way, one may obtain a conclusion which is close to point (c) stated above.
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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 (
Page 32 and 33: 172 Geoffrey Grimmett Proof of Theo
Page 34 and 35: 174 Geoffrey Grimmett 5.3 Site and
Page 36 and 37: 176 Geoffrey Grimmett 6.1 Using Sub
Page 38 and 39: 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 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 90 and 91: 230 Geoffrey Grimmett for crossing
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
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272 Geoffrey Grimmett which, via Le
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274 Geoffrey Grimmett Turning to th
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276 Geoffrey Grimmett Each circuit
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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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