PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

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220 <strong>Geoffrey</strong> Grimmett where α ′ = α ′ (p) is strictly positive and continuous for 0 < p < 1 (and we have used the fact (6.7) that χ(pc) = ∞). � Finally we discuss the verification of the triangle condition T(pc) < ∞. This has been proved for large d (currently d ≥ 19) by Hara and Slade [177, 178, 179, 180, 181], and is believed to hold for d ≥ 7. The corresponding condition for a ‘spread-out’ percolation model, having large but finite-range links rather than nearest-neighbour only, is known to hold for d > 6. The proof that T(pc) < ∞ is long and technical, and is to be found in [179]; since the present author has no significant improvement on that version, the details are not given here. Instead, we survey briefly the structure of the proof. The triangle function (8.9) involves convolutions, and it is therefore natural to introduce the Fourier transform of the connectivity function τp(x, y) = Pp(x ↔ y). More generally, if f : Z d → R is summable, we define �f(θ) = � f(x)e iθ·x , for θ = (θ1, . . . , θd) ∈ [−π, π] d , x∈Z d where θ · x = �d j=1 θjxj. If f is symmetric (i.e., f(x) = f(−x) for all x), then � f is real. We have now that (8.23) T(p) = � τp(0, x)τp(x, y)τp(y, 0) = (2π) −d � �τp(θ) 3 dθ. x,y [−π,π] d The proof that T(pc) < ∞ involves an upper bound on �τp, namely the so called infra-red bound (8.24) �τp(θ) ≤ c(p) |θ| 2 where |θ| = √ θ · θ. It is immediate via (8.23) that the infra-red bound (8.24) implies that T(p) < ∞. Also, if (8.24) holds for some c(p) which is uniformly bounded for p < pc, then T(pc) = limp↑pc T(p) < ∞. It is believed that (8.25) �τp(θ) ≃ 1 |θ| 2−η as |θ| → 0 where η is the critical exponent given in the table of Section 8.2. Theorem 8.26 (Hara–Slade [179]). There exists D satisfying D > 6 such that, if d ≥ D, then �τp(θ) ≤ c(p) |θ| 2
T(p) − 1 or W(p) 4k/d 3k/d Percolation and Disordered Systems 221 3 2d Fig. 8.2. There is a ‘forbidden region’ for the pairs (p, T(p) − 1) and (p, W(p)), namely the shaded region in this figure. The quantity k denotes kT or kW as appropriate. for some c(p) which is uniformly bounded for p < pc. Also T(pc) < ∞. The proof is achieved by establishing and using the following three facts: (a) T(p) and W(p) = � |x| 2 τp(0, x) 2 4 2d x∈Z d are continuous for p ≤ pc; (b) there exist constants kT and kW such that T(p) ≤ 1 + kT d , W(p) ≤ kW d p , for p ≤ 1 2d ; (c) for large d, and for p satisfying (2d) −1 ≤ p < pc, we have that whenever T(p) ≤ 1 + 3kT d T(p) ≤ 1 + 4kT d , W(p) ≤ 3kW d , W(p) ≤ 4kW d , p ≤ 3 2d , p ≤ 4 2d . Fact (a) is a consequence of the continuity of τp and monotone convergence. Fact (b) follows by comparison with a simpler model (the required comparison is successful for sufficiently small p, namely p ≤ (2d) −1 ). Fact (c) is much harder to prove, and it is here that the ‘lace expansion’ is used. Part (c) implies that there is a ‘forbidden region’ for the pairs (p, T(p)) and (p, W(p)); see Figure 8.2. Since T and W are finite for small p, and continuous up to pc, part (c) implies that T(pc) ≤ 1 + 3kT d , W(pc) ≤ 3kW d , pc ≤ 3 2d .
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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
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 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 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
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(
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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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