PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

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302 <strong>Geoffrey</strong> Grimmett 355. Wierman, J. C. (1992). Equality of the bond percolation critical exponents for two pairs of dual lattices. Combinatorics, Probability, Computing 1, 95–105. 356. Wierman, J. C. (1994). Equality of directional critical exponents in multiparameter percolation models. Journal of Physics A: Mathematical and General 27, 1851–1858. 357. Wierman, J. C. (1995). Substitution method critical probability bounds for the square lattice site percolation model. Combinatorics, Probability, Computing 4, 181–188. 358. Wilson, R. J. (1979). Introduction to Graph Theory. Longman, London. 359. Wu, F. Y. (1982). The Potts model. Reviews in Modern Physics 54, 235–268. 360. Yang, W. and Zhang, Y. (1992). A note on differentiability of the cluster density for independent percolation in high dimensions. Journal of Statistical Physics 66, 1123–1138. 361. Zhang, Y. (1991). A power law for connectedness of some random graphs at the critical point. Random Structures and Algorithms 2, 101–119. 362. Zhang, Y. (1992). Failure of the power laws on some subgraphs of the Z 2 lattice. Journal of Physics A: Mathematical and General 25, 6617– 6622. 363. Zhang, Y. (1993). A shape theorem for epidemics and forest fires with finite range interactions. Annals of Probability 21, 1755–1781. 364. Zhang, Y. (1994). A note on inhomogeneous percolation. Annals of Probability 22, 803–820. 365. Zhang, Y. (1994). Analyticity properties at the supercritical state, preprint. 366. Zhang, Y. (1995). The fractal volume of the two-dimensional invasion percolation cluster. Communications in Mathematical Physics 167, 237–254. 367. Zhang, Y. (1995). Supercritical behaviors in first-passage percolation. Stochastic Processes and their Applications 59, 251–266. 368. Zhang, Y. (1995). A limit theorem for matching random sequences allowing deletions. Annals of Applied Probability 5, 1236–1240. 369. Zhang, Y. (1996). The complete convergence theorem on trees. Annals of Probability 24, 1408–1443. 370. Zhang, Y. (1996). Continuity of percolation probability in ∞ + 1 dimensions. Journal of Applied Probability 33, 427–433. 371. Zhang, Y. (1996). Two critical behaviors of first passage time, preprint. 372. Zhang, Y. (1996). Some power laws on two dimensional critical bond percolation, preprint. 373. Zhang, Y. (1996). Divergence of the bulk resistance at criticality in disordered media. Journal of Statistical Physics 84, 263–267.
Percolation and Disordered Systems 303 374. Zhang, Y. and Zhang, Y. C. (1984). A limit theorem for N0n/n in first passage percolation. Annals of Probability 12, 1068–1076. 375. Ziff, R. M., Kong, X. P., and Cohen, E. G. D. (1991). Lorentz latticegas and kinetic-walk model. Physical Review A 44, 2410–2428.
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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
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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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244 Geoffrey Grimmett Fig. 10.7. 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
Page 112 and 113: 252 Geoffrey Grimmett Fig. 11.1. Th
Page 114 and 115: 254 Geoffrey Grimmett Fig. 11.2. Th
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 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.
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