PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

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290 <strong>Geoffrey</strong> Grimmett 158. Grimmett, G. R. (1994). The random-cluster model. Probability, Statistics and Optimisation (F. P. Kelly, ed.), John Wiley & Sons, Chichester, pp. 49–63. 159. Grimmett, G. R. (1994). Percolative problems. Probability and Phase Transition (G. R. Grimmett, ed.), Kluwer, Dordrecht, pp. 69–86. 160. Grimmett, G. R. (1994). Probability and Phase Transition. editor. Kluwer, Dordrecht. 161. Grimmett, G. R. (1995). Comparison and disjoint-occurrence inequalities for random-cluster models. Journal of Statistical Physics 78, 1311–1324. 162. Grimmett, G. R. (1995). The stochastic random-cluster process and the uniqueness of random-cluster measures. Annals of Probability 23, 1461–1510. 163. Grimmett, G. R., Kesten, H., and Zhang, Y. (1993). Random walk on the infinite cluster of the percolation model. Probability Theory and Related Fields 96, 33–44. 164. Grimmett, G. R. and Marstrand, J. M. (1990). The supercritical phase of percolation is well behaved. Proceedings of the Royal Society (London), Series A 430, 439–457. 165. Grimmett, G. R., Menshikov, M. V., and Volkov, S. E. (1996). Random walks in random labyrinths. Markov Processes and Related Fields 2, 69–86. 166. Grimmett, G. R. and Newman, C. M. (1990). Percolation in ∞ + 1 dimensions. Disorder in Physical Systems (G. R. Grimmett and D. J. A. Welsh, eds.), Clarendon Press, Oxford, pp. 167–190. 167. Grimmett, G. R. and Piza, M. S. T. (1997). Decay of correlations in subcritical Potts and random-cluster models. Communications in Mathematical Physics 189, 465–480. 168. Grimmett, G. R. and Stacey, A. M. (1998). Inequalities for critical probabilities of site and bond percolation. Annals of Probability 26, 1788–1812. 169. Grimmett, G. R. and Stirzaker, D. R. (1992). Probability and Random Processes. Oxford University Press, Oxford. 170. Häggström, O. (1996). The random-cluster model on a homogeneous tree. Probability Theory and Related Fields 104, 231–253. 171. Häggström, O. and Meester, R. (1995). Asymptotic shapes in stationary first passage percolation. Annals of Probability 23, 1511– 1522. 172. Häggström, O. and Meester, R. (1996). Nearest neighbour and hard sphere models in continuum percolation. Random Structures and Algorithms, 295–315. 173. Häggström, O. and Pemantle, R. (1998). First-passage percolation and a model for competing spatial growth. Journal of Applied Probability, 683–692.
Percolation and Disordered Systems 291 174. Häggström, O., Peres, Y., and Steif, J. (1997). Dynamical percolation. Annales de l’Institut Henri Poincaré: Probabilités et Statistiques 33, 497–528. 175. Hammersley, J. M. (1957). Percolation processes. Lower bounds for the critical probability. Annals of Mathematical Statistics 28, 790–795. 176. Hammersley, J. M. (1959). Bornes supérieures de la probabilité critique dans un processus de filtration. Le Calcul de Probabilités et ses Applications, CNRS, Paris, pp. 17–37. 177. Hara, T. and Slade, G. (1989). The mean-field critical behaviour of percolation in high dimensions. Proceedings of the IXth International Congress on Mathematical Physics (B. Simon, A. Truman, I. M. Davies, eds.), Adam Hilger, Bristol, pp. 450–453. 178. Hara, T. and Slade, G. (1989). The triangle condition for percolation. Bulletin of the American Mathematical Society 21, 269–273. 179. Hara, T. and Slade, G. (1990). Mean-field critical behaviour for percolation in high dimensions. Communications in Mathematical Physics 128, 333–391. 180. Hara, T. and Slade, G. (1994). Mean-field behaviour and the lace expansion. Probability and Phase Transition (G. R. Grimmett, ed.), Kluwer, Dordrecht, pp. 87–122. 181. Hara, T. and Slade, G. (1995). The self-avoiding walk and percolation critical points in high dimensions. Combinatorics, Probability, Computing 4, 197–215. 182. Harris, T. E. (1960). A lower bound for the critical probability in a certain percolation process. Proceedings of the Cambridge Philosophical Society 56, 13–20. 183. Hauge, E. H. and Cohen, E. G. D. (1967). Normal and abnormal effects in Ehrenfest wind-tree model. Physics Letters 25A, 78–79. 184. Hawkes, J. (1981). Trees generated by a simple branching process. Journal of the London Mathematical Society 24, 373–384. 185. Higuchi, Y. (1989). A remark on the percolation for the 2D Ising model. Osaka Journal of Mathematics 26, 207–224. 186. Higuchi, Y. (1991). Level set representation for the Gibbs state of the ferromagnetic Ising model. Probability Theory and Related Fields 90, 203–221. 187. Higuchi, Y. (1992). Percolation (in Japanese). Yuseisha/Seiunsha. 188. Higuchi, Y. (1993). A sharp transition for the two-dimensional Ising percolation. Probability Theory and Related Fields 97, 489–514. 189. Higuchi, Y. (1993). Coexistence of infinite *-clusters II: Ising percolation in two dimensions. Probability Theory and Related Fields 97, 1–34. 190. Higuchi, Y. (1996). Ising model percolation. Sugaku Expositions, vol. 47, pp. 111–124.
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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 ′
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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
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(
Page 130 and 131: 270 Geoffrey Grimmett That is to sa
Page 132 and 133: 272 Geoffrey Grimmett which, via Le
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 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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