PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

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272 <strong>Geoffrey</strong> Grimmett which, via Lemma 13.22, implies as in (13.25) that φs,q(Hi) ≥ D(log n)2 (log log n) ∆4 for some D > 0. By (13.28), and the fact that K = ⌊n/m⌋, φs,q(Fn) ≥ D ′ n (log log n) ∆4 for some D ′ > 0. By (13.24), � c5n φr,q(An) ≤ exp − (loglog n) ∆4 Since ∆4 > 1, this implies the claim of the theorem with k = 1. The claim for general k requires k − 1 further iterations of the argument. (b) We omit the proof of this part. The fundamental argument is taken from [138], and the details are presented in [167]. � 13.5 The Case of Two Dimensions In this section we consider the case of random-cluster measures on the square lattice L 2 . Such measures have a property of self-duality which generalises that of bond percolation. We begin by describing this duality. Let G = (V, E) be a plane graph with planar dual G d = (V d , E d ). Any configuration ω ∈ ΩE gives rise to a dual configuration ω d ∈ Ω E d defined as follows. If e (∈ E) is crossed by the dual edge e d (∈ E d ), we define ω d (e d ) = 1−ω(e). As usual, η(ω) denotes the set {e : ω(e) = 1} of edges which are open in ω. By drawing a picture, one may be convinced that every face of (V, η(ω)) contains a unique component of (V d , η(ω d )), and therefore the number f(ω) of faces (including the infinite face) of (V, η(ω)) satisfies f(ω) = k(ω d ). See Figure 13.1. (Note that this definition of the dual configuration differs slightly from that used earlier for two-dimensional percolation.) The random-cluster measure on G is given by Using Euler’s formula, φG,p,q(ω) ∝ � . � � |η(ω)| p q 1 − p k(ω) . k(ω) = |V | − |η(ω)| + f(ω) − 1, and the facts that f(ω) = k(ω d ) and |η(ω)| + |η(ω d )| = |E|, we have that φG,p,q(ω) ∝ � q(1 − p) p � |η(ω d )| q k(ωd ) ,
Percolation and Disordered Systems 273 Fig. 13.1. A primal configuration ω (with solid lines) and its dual configuration ω d (with dashed lines). The arrows join the given vertices of the dual to a dual vertex in the infinite face. Note that each face of the primal graph (including the infinite face) contains a unique component of the dual graph. which is to say that (13.32) φG,p,q(ω) = φ G d ,p d ,q(ω d ) for ω ∈ ΩE, where the dual parameter p d is given according to (13.33) pd q(1 − p) = . 1 − pd p The unique fixed point of the mapping p ↦→ pd is easily seen to be given by p = κq where √ q κq = 1 + √ q . If we keep track of the constants of proportionality in the above calculation, we find that the partition function ZG,p,q = � satisfies the duality relation ω∈ΩE |V |−1 (13.34) ZG,p,q = q which, when p = p d = κq, becomes p |η(ω)| (1 − p) |E\η(ω)| q k(ω) � 1 − p p d � |E| Z G d ,p d ,q |V |−1−1 (13.35) ZG,κq,q = q 2 |E| ZGd ,κq,q. We shall find a use for this later.
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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 ′
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
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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
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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 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 (
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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