PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

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210 <strong>Geoffrey</strong> Grimmett Fig. 7.10. Illustrations of the block construction for the proof of Theorem 7.35 presented in [49]. The grey regions contain open paths joining the black ‘inlet’ to the three ‘outlets’. The fundamental building block is a rectangle rather than a square. We have no control of the aspect ratio of this rectangle, and consequently two cases with somewhat different geometries need to be considered. Compare with Figure 7.7.
or Percolation and Disordered Systems 211 θ ′ (π) + θ(π) ≥ 1. Integrate this over the interval (pc, p) to obtain θ(p)e p − θ(pc)e pc ≥ e p − e pc , pc ≤ p, whence it is an easy exercise to show that (7.36) θ(p) − θ(pc) ≥ a(p − pc), pc ≤ p, for some positive constant a. The above argument may be made rigorous. Differential inequalities of the type above are used widely in percolation and disordered systems. 7.6 Cluster-Size Distribution When p < pc, the tail of the cluster-size |C| decays exponentially. Exponential decay is not correct when p > pc, but rather ‘stretched exponential decay’. Theorem 7.37. Suppose pc < p < 1. There exist positive constants α(p), β(p) such that, for all n, (7.38) exp � −αn (d−1)/d� ≤ Pp(|C| = n) ≤ exp � −βn (d−1)/d� . See [G] for a proof of this theorem. The reason for the power n (d−1)/d is roughly as follows. It is thought that a large finite cluster is most likely created as a cluster of compact shape, all of whose boundary edges are closed. Now, if a ball has volume n, then its surface area has order n (d−1)/d . The price paid for having a surface all of whose edges are closed is (1 −p) m where m is the number of such edges. By the above remark, m should have order n (d−1)/d , as required for (7.38). It is believed that the limit � 1 (7.39) γ(p) = lim − n→∞ n (d−1)/d log Pp(|C| � = n) exists, but no proof is known. Much more is known in two dimensions than for general d. The size and geometry of large finite clusters have been studied in detail in [37], where it was shown that such clusters may be approximated by the so called ‘Wulff shape’. This work includes a proof of the existence of the limit in (7.39) when d = 2.
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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]
Page 20 and 21: 160 Geoffrey Grimmett whence d dp P
Page 22 and 23: 162 Geoffrey Grimmett Fig. 4.1. An
Page 24 and 25: 164 Geoffrey Grimmett s 1 θ = 0 pc
Page 26 and 27: 166 Geoffrey Grimmett ∂B(n) 0 e f
Page 28 and 29: 168 Geoffrey Grimmett Fig. 4.5. A s
Page 30 and 31: 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 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 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
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260 Geoffrey Grimmett (b) Under wha
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262 Geoffrey Grimmett is obtained b
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264 Geoffrey Grimmett 13.2 Weak Lim
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266 Geoffrey Grimmett where p = 1
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268 Geoffrey Grimmett Evidently pg(
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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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