PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

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278 <strong>Geoffrey</strong> Grimmett If A(q) < 1 (which holds for sufficiently large q), then φ 1 B,p,q(0 ↔ ∂B) = φ 1 � � B,p,q OC(Γ) occurs for no Γ ≥ 1 − A(q) > 0. (We have used the assumption of wired boundary conditions here.) This implies, by taking the limit n → ∞, that θ1 (p, q) > 0 when p = √ q/(1 + √ q). Using (13.39), this implies parts (a) and (b) of the theorem, when q is sufficiently large. For general q > Q, we have only that A(q) < ∞. In this case, we find N (< n) such that � Γ outside B(N) φ 1 � � 1 B,p,q OC(Γ) < 2 where Γ is said to be outside B(N) if it contains B(N) in its interior. This implies that φ1 � � 1 B,p,q B(N) ↔ ∂B ≥ 2 . Let n → ∞, and deduce that φ1 � � 1 p,q B(N) ↔ ∞ ≥ 2 , implying that θ1 (p, q) > 0 as required. Turning to part (c) 12 , let p = pd = √ q/(1 + √ q) and n ≤ r. Let An be the (cylinder) event that the point ( 1 1 2 , 2 ) lies in the interior of an open circuit of length at least n, this circuit having the property that its interior is contained in the interior of no open circuit having length strictly less than n. We have from (13.36) and (13.45) that (13.46) φ 0 B(r),p,q (An) ≤ ∞� m=n mam q � q (1 + √ q) 4 � m/4 for all large r. [Here, we use the observation that, if An occurs in B(r), then there exists a maximal open circuit Γ of B(r) containing ( 1 1 2 , 2 ). In the dual of B(r), Γ constitutes an outer circuit.] We write LRn for the event that there is an open crossing of the rectangle Rn = [0, n] ×[0, 2n] from its left to its right side, and we set λn = φ0 p,q (LRn). We may find a point x on the left side of Rn and a point y on the right side such that φ 0 p,q (x ↔ y in Rn) λn ≥ . (2n + 1) 2 By placing six of these rectangles side by side (as in Figure 13.4), we find by the FKG inequality that (13.47) φ 0 p,q � � � x ↔ x + (6n, 0) in [0, 6n] × [0, 2n] ≥ λn (2n + 1) 2 � 6 . 12 The present proof was completed following a contribution by Ken Alexander, see [36] for related material.
x Percolation and Disordered Systems 279 y x + (6n, 0) Fig. 13.4. Six copies of a rectangle having width n and height 2n may be put together to make a rectangle with size 6n by 2n. If each is crossed by an open path joining the images of x and y, then the larger rectangle is crossed between its shorter sides. Fig. 13.5. If each of four rectangles having dimensions 6n by 2n is crossed by an open path between its shorter sides, then the annulus contains an open circuit having the origin in its interior. We now use four copies of the rectangle [0, 6n]×[0, 2n] to construct an annulus around the origin (see Figure 13.5). If each of these copies contains an open crossing, then the annulus contains a circuit. Using the FKG inequality again, we deduce that (13.48) φ 0 � p,q(A4n) ≥ λn (2n + 1) 2 � 24 . Finally, if 0 ↔ ∂B(n), then one of the four rectangles [0, n] × [−n, n], [−n, n] × [0, n], [−n, 0] × [−n, n], [−n, n] × [−n, 0] is traversed by an open path betwen its two longer sides. This implies that (13.49) φ 0 � � p,q 0 ↔ ∂B(n) ≤ 4λn. Combining (13.46)–(13.49), we obtain that φ 0 � � � p,q 0 ↔ ∂B(n) ≤ 4 (2n + 1) 2 φ 0 p,q (A4n) �1/24 � ≤ 4 (2n + 1) 2 ∞� � mam q q (1 + √ q) 4 � � m/4 1/24 . m=4n As before, m −1 log am → µ as m → ∞, whence part (c) follows. �
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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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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
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180 Geoffrey Grimmett so that (3.10
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182 Geoffrey Grimmett x 2 y 2 x 4 x
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184 Geoffrey Grimmett Similarly, Pp
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186 Geoffrey Grimmett It remains to
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188 Geoffrey Grimmett Fix β < pc a
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190 Geoffrey Grimmett where δ 2 =
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192 Geoffrey Grimmett Fig. 7.1. Tak
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196 Geoffrey Grimmett 4. The open c
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198 Geoffrey Grimmett For n > m, le
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200 Geoffrey Grimmett Lemma 7.17. I
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202 Geoffrey Grimmett Lemma 7.24. S
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204 Geoffrey Grimmett 000 111 000 1
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208 Geoffrey Grimmett Fig. 7.9. The
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210 Geoffrey Grimmett Fig. 7.10. Il
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214 Geoffrey Grimmett may be shown
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216 Geoffrey Grimmett Lemma 8.11. L
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218 Geoffrey Grimmett 0 u noting th
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220 Geoffrey Grimmett where α ′
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222 Geoffrey Grimmett The infra-red
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224 Geoffrey Grimmett 9. PERCOLATIO
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226 Geoffrey Grimmett Fig. 9.2. If
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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
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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 140 and 141: 280 Geoffrey Grimmett REFERENCES 1.
Page 142 and 143: 282 Geoffrey Grimmett 30. Alexander
Page 144 and 145: 284 Geoffrey Grimmett 63. Berg, J.
Page 146 and 147: 286 Geoffrey Grimmett 93. Chayes, J
Page 148 and 149: 288 Geoffrey Grimmett 123. Durrett,
Page 150 and 151: 290 Geoffrey Grimmett 158. Grimmett
Page 152 and 153: 292 Geoffrey Grimmett 191. Higuchi,
Page 154 and 155: 294 Geoffrey Grimmett 224. Koteck´
Page 156 and 157: 296 Geoffrey Grimmett 259. Meester,
Page 158 and 159: 298 Geoffrey Grimmett 290. Newman,
Page 160 and 161: 300 Geoffrey Grimmett 323. Roy, R.
Page 162 and 163: 302 Geoffrey Grimmett 355. Wierman,
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PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

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