PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

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288 <strong>Geoffrey</strong> Grimmett 123. Durrett, R. T. and Schonmann, R. H. (1988). The contact process on a finite set II. Annals of Probability 16, 1570–1583. 124. Durrett, R. T. and Schonmann, R. H. (1988). Large deviations for the contact process and two dimensional percolation. Probability Theory and Related Fields 77, 583–603. 125. Durrett, R. T., Schonmann, R. H., and Tanaka, N. I. (1989). The contact process on a finite set III. The critical case. Annals of Probability 17, 1303–1321. 126. Durrett, R. T., Schonmann, R. H., and Tanaka, N. I. (1989). Correlation lengths for oriented percolation. Journal of Statistical Physics 55, 965–979. 127. Edwards, R. G. and Sokal, A. D. (1988). Generalization of the Fortuin–Kasteleyn–Swendsen–Wang representation and Monte Carlo algorithm. The Physical Review D 38, 2009–2012. 128. Ehrenfest, P. (1959). Collected Scientific Papers (M. J. Klein, ed.). North-Holland, Amsterdam. 129. Enter, A. C. D. van and Hollander, F. den (1993). Interacting particle systems and Gibbs measures, Lecture notes for Master Class 1992– 1993, Mathematical Research Institute, The Netherlands. 130. Falconer, K. J. (1986). Random fractals. Mathematical Proceedings of the Cambridge Philosophical Society 100, 559–582. 131. Falconer, K. J. (1986). Sets with prescribed projections and Nikodym sets. Proceedings of the London Mathematical Society 53, 48–64. 132. Falconer, K. J. (1989). Projections of random Cantor sets. Journal of Theoretical Probability 2, 65–70. 133. Falconer, K. J. and Grimmett, G. R. (1992). On the geometry of random Cantor sets and fractal percolation. Journal of Theoretical Probability 5, 465–485; Erratum (1994) 7, 209–210. 134. Fernández, R., Fröhlich, J., and Sokal, A. D. (1992). Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory. Springer-Verlag, Berlin. 135. Fill, J. and Pemantle, R. (1992). Oriented percolation, first-passage percolation and covering times for Richardson’s model on the n-cube. Annals of Applied Probability 3, 593–629. 136. Fiocco, M. and Zwet, W. R. van (1996). Parameter estimation for the supercritical contact process. Bernoulli 9, 1071–1092. 137. Fisher, M. E. (1961). Critical probabilities for cluster size and percolation problems. Journal of Mathematical Physics 2, 620–627. 138. Fontes, L. and Newman, C. M. (1993). First passage percolation for random colorings of Z d . Annals of Applied Probability 3, 746–762; Erratum 4, 254. 139. Fortuin, C. M. (1971). On the random-cluster model. Doctoral thesis, University of Leiden. 140. Fortuin, C. M. (1972). On the random cluster model. II. The percolation model. Physica 58, 393–418.
Percolation and Disordered Systems 289 141. Fortuin, C. M. (1972). On the random cluster model. III. The simple random-cluster process. Physica 59, 545–570. 142. Fortuin, C. M. and Kasteleyn, P. W. (1972). On the random cluster model. I. Introduction and relation to other models. Physica 57, 536– 564. 143. Fortuin, C. M., Kasteleyn, P. W., and Ginibre, J. (1971). Correlation inequalities on some partially ordered sets. Communications in Mathematical Physics 22, 89–103. 144. Friedgut, E. and Kalai, G. (1996). Every monotone graph property has a sharp threshold. Proceedings of the American Mathematical Society 124, 2993–3002. 145. Gandolfi, A., Grimmett, G. R., and Russo, L. (1988). On the uniqueness of the infinite open cluster in the percolation model. Communications in Mathematical Physics 114, 549–552. 146. Gandolfi, A., Keane, M., and Newman, C. M. (1992). Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Probability Theory and Related Fields 92, 511–527. 147. Gandolfi, A. and Kesten, H. (1994). Greedy lattice animals II: Linear growth. Advances in Applied Probability 4, 76–107. 148. Gandolfi, A., Newman, C. M., and Stein, D. L. (1993). Exotic states in long range spin glasses. Communications in Mathematical Physics 157, 371–387. 149. Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. Walter de Gruyter, Berlin. 150. Gordon, D. M. (1991). Percolation in high dimensions. Bulletin of the London Mathematical Society 44, 373–384. 151. Graf, S. (1987). Statistically self-similar fractals. Probability Theory and Related Fields 74, 357–392. 152. Graf, S., Mauldin, R. D., and Williams, S. C. (1987). Exact Hausdorff dimension in random recursive constructions. Memoirs of the American Mathematical Society 381. 153. Griffeath, D. (1979). Additive and Cancellative Interacting Particle Systems. Lecture Notes in Mathematics, vol. 724. Springer, Berlin. 154. Grimmett, G. R. (1978). The rank functions of large random lattices. Journal of London Mathematical Society 18, 567–575. 155. Grimmett, G. R. (1989). Percolation. Springer-Verlag, Berlin. 156. Grimmett, G. R. (1993). Differential inequalities for Potts and random-cluster processes. Cellular Automata and Cooperative Systems (N. Boccara et al., eds.), Kluwer, Dordrecht, pp. 227–236. 157. Grimmett, G. R. (1994). Potts models and random-cluster processes with many-body interactions. Journal of Statistical Physics 75, 67– 121.
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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
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
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 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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