PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

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192 <strong>Geoffrey</strong> Grimmett Fig. 7.1. Take a box B which intersects at least three distinct infinite open clusters, and then alter the configuration inside B in order to create a configuration in which 0 is a trifurcation. least three distinct infinite clusters with probability at least 1 2 . By changing the configuration inside B(n), we may turn the origin into a trifurcation (see Figure 7.1). The corresponding set of configurations has strictly positive probability, so that Pp(T0) > 0 in (7.4). Before turning to the geometry, we present a lemma concerning partitions. Let Y be a finite set with |Y | ≥ 3, and define a 3-partition Π = {P1, P2, P3} of Y to be a partition of Y into exactly three non-empty sets P1, P2, P3. }, we say that Π For 3-partitions Π = {P1, P2, P3} and Π ′ = {P ′ 1 , P ′ 2 , P ′ 3 and Π ′ are compatible if there exists an ordering of their elements such that P1 ⊇ P ′ 2 ∪ P ′ 3 (or, equivalently, that P ′ 1 ⊇ P2 ∪ P3). A collection P of 3partitions is compatible if each pair therein is compatible. Lemma 7.5. If P is a compatible family of distinct 3-partitions of Y , then |P| ≤ |Y | − 2. Proof. There are several ways of doing this; see [82]. For any set Q of distinct compatible 3-partitions of Y , we define an equivalence relation ∼ on Y by x ∼ y if, for all Π ∈ Q, x and y lie in the same element of Π. Write α(Q) for the number of equivalence classes of ∼. Now, write P = (Π1, Π2, . . . , Πm) in some order, and let αk = α(Π1, Π2, . . . , Πk). Evidently α1 = 3 and, using the compatibility of Π1 and Π2, we have that α2 ≥ 4. By comparing Πr+1 with Π1, Π2, . . . , Πr in turn, and using their compatibility, one sees that α(Π1, Π2, . . . , Πr+1) ≥ α(Π1, Π2, . . . , Πr) + 1, whence αm ≥ α1 + (m − 1) = m + 2. However αm ≤ |Y |, and the claim of the lemma follows. � Let K be a connected open cluster of B(n), and write ∂K = K ∩ ∂B(n). If x (∈ B(n − 1)) is a trifurcation in K, then the removal of x induces a 3-partition ΠK(x) = {P1, P2, P3} of ∂K with the properties that (a) Pi is non-empty, for i = 1, 2, 3, (b) Pi is a subset of a connected subgraph of B(n)\{x}, (c) Pi � Pj in B(n), if i �= j.
Percolation and Disordered Systems 193 Fig. 7.2. Two trifurcations x and x ′ belonging to a cluster K of B(n). They induce compatible partitions of ∂K. Furthermore, if x and x ′ are distinct trifurcations of K ∩ B(n − 1), then ΠK(x) and ΠK(x ′ ) are distinct and compatible; see Figure 7.2. It follows by Lemma 7.5 that the number T(K) of trifurcations in K ∩ B(n − 1) satisfies T(K) ≤ |∂K| − 2. We sum this inequality over all connected clusters of B(n), to obtain that � x∈B(n−1) x x ′ 1Tx ≤ |∂B(n)|. Take expectations, and use (7.4) to find that |B(n − 1)|Pp(T0) ≤ |∂B(n)|, which is impossible for large n since the left side grows as n d and the right side as n d−1 . This contradiction completes the proof. �
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Page 4 and 5: 144 Geoffrey Grimmett ÈÊÇ��
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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 24 and 25: 164 Geoffrey Grimmett s 1 θ = 0 pc
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Page 32 and 33: 172 Geoffrey Grimmett Proof of Theo
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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
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Page 56 and 57: 196 Geoffrey Grimmett 4. The open c
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Page 60 and 61: 200 Geoffrey Grimmett Lemma 7.17. I
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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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