PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

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148 <strong>Geoffrey</strong> Grimmett θ(p) 1 pc 1 p Fig. 1.1. It is generally believed that the percolation probability θ(p) behaves roughly as indicated here. It is known, for example, that θ is infinitely differentiable except at the critical point pc. The possibility of a jump discontinuity at pc has not been ruled out when d ≥ 3 but d is not too large. 1.2 Some Possible Questions Here are some apparently reasonable questions, some of which turn out to be feasible. • What is the value of pc? • What are the structures of the subcritical and supercritical phases? • What happens when p is near to pc? • Are there other points of phase transition? • What are the properties of other ‘macroscopic’ quantities, such as the mean size of the open cluster containing the origin? • What is the relevance of the choice of dimension or lattice? • In what ways are the large-scale properties different if the states of nearby edges are allowed to be dependent rather than independent? There is a variety of reasons for the explosion of interest in the percolation model, and we mention next a few of these. • The problems are simple and elegant to state, and apparently hard to solve. • Their solutions require a mixture of new ideas, from analysis, geometry, and discrete mathematics. • Physical intuition has provided a bunch of beautiful conjectures. • Techniques developed for percolation have applications to other more complicated spatial random processes, such as epidemic models. • Percolation gives insight and method for understanding other physical models of spatial interaction, such as Ising and Potts models. • Percolation provides a ‘simple’ model for porous bodies and other ‘transport’ problems.
Percolation and Disordered Systems 149 The rate of publication of papers on percolation and its ramifications is very high in the physics journals, although substantial mathematical contributions are rare. The depth of the ‘culture chasm’ is such that few (if anyone) can honestly boast to understand all the major mathematical and physical ideas which have contributed to the subject. 1.3 History In 1957, Simon Broadbent and John Hammersley [80] presented a model for a disordered porous medium which they called the percolation model. Their motivation was perhaps to understand flow through a discrete disordered system, such as particles flowing through the filter of a gas mask, or fluid seeping through the interstices of a porous stone. They proved in [80, 175, 176] that the percolation model has a phase transition, and they developed some technology for studying the two phases of the process. These early papers were followed swiftly by a small number of high quality articles by others, particularly [137, 182, 338], but interest flagged for a period beginning around 1964. Despite certain appearances to the contrary, some individuals realised that a certain famous conjecture remained unproven, namely that the critical probability of bond percolation on the square lattice equals 1 2 . Fundamental rigorous progress towards this conjecture was made around 1976 by Russo [325] and Seymour and Welsh [337], and the conjecture was finally resolved in a famous paper of Kesten [201]. This was achieved by a development of a sophisticated mechanism for studying percolation in two dimensions, relying in part on path-intersection properties which are not valid in higher dimensions. This mechanism was laid out more fully by Kesten in his monograph [203]. Percolation became a subject of vigorous research by mathematicians and physicists, each group working in its own vernacular. The decade beginning in 1980 saw the rigorous resolution of many substantial difficulties, and the formulation of concrete hypotheses concerning the nature of phase transition. The principal progress was on three fronts. Initially mathematicians concentrated on the ‘subcritical phase’, when the density p of open edges satisfies p < pc (here and later, pc denotes the critical probability). It was in this context that the correct generalisation of Kesten’s theorem was discovered, valid for all dimensions (i.e., two or more). This was achieved independently by Aizenman and Barsky [12] and Menshikov [267, 268]. The second front concerned the ‘supercritical phase’, when p > pc. The key question here was resolved by Grimmett and Marstrand [164] following work of Barsky, Grimmett, and Newman [49]. The critical case, when p is near or equal to the critical probability pc, remains largely unresolved by mathematicians (except when d is sufficiently large). Progress has certainly been made, but we seem far from understanding the beautiful picture of the phase transition, involving scaling theory and renormalisation, which is displayed before us by physicists. This multifaceted
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Page 20 and 21: 160 Geoffrey Grimmett whence d dp P
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Page 24 and 25: 164 Geoffrey Grimmett s 1 θ = 0 pc
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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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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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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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