PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

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144 <strong>Geoffrey</strong> Grimmett ÈÊÇ��ÁÄÁËÌ�, s. m. (Gram. Théol.) celui qui tient pour la doctrine abominable des opinions rendues probables par la décision d’un casuiste, & qui assure l’innocence de l’action faite en conséquence. Pascal a foudroyé ce systême, qui ouvroit la porte au crime en accordant à l’autorité les prérogatives de la certitude, à l’opinion & la sécurité qui n’appartient qu’à la bonne conscience. This work was aided by partial financial support from the European Union under contract CHRX–CT93–0411, and from the Engineering and Physical Sciences Research Council under grant GR/L15425. <strong>Geoffrey</strong> R. Grimmett Statistical Laboratory University of Cambridge 16 Mill Lane Cambridge CB2 1SB United Kingdom. October 1996 Note added at reprinting. GRG thanks Roberto Schonmann for pointing out an error in the proof of equation (8.20), corrected in this reprint. June 2012
Percolation and Disordered Systems 145 CONTENTS 1. Introductory Remarks 147 1.1 Percolation 147 1.2 Some possible questions 148 1.3 History 149 2. Notation and Definitions 151 2.1 Graph terminology 151 2.2 Probability 152 2.3 Geometry 153 2.4 A partial order 154 2.5 Site percolation 155 3. Phase Transition 156 3.1 Percolation probability 156 3.2 Existence of phase transition 156 3.3 A question 158 4. Inequalities for Critical Probabilities 159 4.1 Russo’s formula 159 4.2 Strict inequalities for critical probabilities 161 4.3 The square and triangular lattices 162 4.4 Enhancements 167 5. Correlation Inequalities 169 5.1 FKG inequality 169 5.2 Disjoint occurrence 172 5.3 Site and bond percolation 174 6. Subcritical Percolation 176 6.1 Using subadditivity 176 6.2 Exponential decay 179 6.3 Ornstein–Zernike decay 190 7. Supercritical Percolation 191 7.1 Uniqueness of the infinite cluster 191 7.2 Percolation in slabs 194 7.3 Limit of slab critical points 195 7.4 Percolation in half-spaces 209 7.5 Percolation probability 209 7.6 Cluster-size distribution 211
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Page 6 and 7: 146 Geoffrey Grimmett 8. Critical P
Page 8 and 9: 148 Geoffrey Grimmett θ(p) 1 pc 1
Page 10 and 11: 150 Geoffrey Grimmett physical imag
Page 12 and 13: 152 Geoffrey Grimmett Fig. 2.1. Par
Page 14 and 15: 154 Geoffrey Grimmett Fig. 2.2. An
Page 16 and 17: 156 Geoffrey Grimmett 3.1 Percolati
Page 18 and 19: 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 =
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194 Geoffrey Grimmett 7.2 Percolati
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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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208 Geoffrey Grimmett Fig. 7.9. The
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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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