PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

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146 <strong>Geoffrey</strong> Grimmett 8. Critical Percolation 212 8.1 Percolation probability 212 8.2 Critical exponents 212 8.3 Scaling theory 212 8.4 Rigorous results 214 8.5 Mean-field theory 214 9. Percolation in Two Dimensions 224 9.1 The critical probability is 1 2 9.2 RSW technology 224 225 9.3 Conformal invariance 228 10. Random Walks in Random Labyrinths 232 10.1 Random walk on the infinite percolation cluster 232 10.2 Random walks in two-dimensional labyrinths 236 10.3 General labyrinths 245 11. Fractal Percolation 251 11.1 Random fractals 251 11.2 Percolation 253 11.3 A morphology 255 11.4 Relationship to Brownian Motion 257 12. Ising and Potts Models 259 12.1 Ising model for ferromagnets 259 12.2 Potts models 260 12.3 Random-cluster models 261 13. Random-Cluster Models 263 13.1 Basic properties 263 13.2 Weak limits and phase transitions 264 13.3 First and second order transitions 266 13.4 Exponential decay in the subcritical phase 267 13.5 The case of two dimensions 272 References 280
1.1. Percolation Percolation and Disordered Systems 147 1. INTRODUCTORY REMARKS We will focus our ideas on a specific percolation process, namely ‘bond percolation on the cubic lattice’, defined as follows. Let L d = (Z d , E d ) be the hypercubic lattice in d dimensions, where d ≥ 2. Each edge of L d is declared open with probability p, and closed otherwise. Different edges are given independent designations. We think of an open edge as being open to the transmission of disease, or to the passage of water. Now concentrate on the set of open edges, a random set. Percolation theory is concerned with ascertaining properties of this set. The following question is considered central. If water is supplied at the origin, and flows along the open edges only, can it reach infinitely many vertices with strictly positive probability? It turns out that the answer is no for small p, and yes for large p. There is a critical probability pc dividing these two phases. Percolation theory is particularly concerned with understanding the geometry of open edges in the subcritical phase (when p < pc), the supercritical phase (when p > pc), and when p is near or equal to pc (the critical case). As an illustration of the concrete problems of percolation, consider the function θ(p), defined as the probability that the origin lies in an infinite cluster of open edges (this is the probability referred to above, in the discussion of pc). It is believed that θ has the general appearance sketched in Figure 1.1. • θ should be smooth on (pc, 1). It is known to be infinitely differentiable, but there is no proof known that it is real analytic for all d. • Presumably θ is continuous at pc. No proof is known which is valid for all d. • Perhaps θ is concave on (pc, 1], or at least on (pc, pc + δ) for some positive δ. • As p ↓ pc, perhaps θ(p) ∼ a(p − pc) β for some constant a and some ‘critical exponent’ β. We stress that, although each of the points raised above is unproved in general, there are special arguments which answer some of them when either d = 2 or d is sufficiently large. The case d = 3 is a good one on which to concentrate.
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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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206 Geoffrey Grimmett Fig. 7.7. An
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210 Geoffrey Grimmett Fig. 7.10. Il
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212 Geoffrey Grimmett 8.1 Percolati
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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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228 Geoffrey Grimmett x Fig. 9.3. I
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230 Geoffrey Grimmett for crossing
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232 Geoffrey Grimmett 10. RANDOM WA
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234 Geoffrey Grimmett the volume of
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236 Geoffrey Grimmett We define sim
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238 Geoffrey Grimmett We now use th
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240 Geoffrey Grimmett p+ 1 pc(site)
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242 Geoffrey Grimmett Fig. 10.6. Th
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246 Geoffrey Grimmett A L d -path i
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248 Geoffrey Grimmett where pc(site
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250 Geoffrey Grimmett now make two
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256 Geoffrey Grimmett II. P(C �=
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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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