PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

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174 <strong>Geoffrey</strong> Grimmett 5.3 Site and Bond Percolation Let G = (V, E) be an infinite connected graph with maximum vertex degree ∆. For a vertex x, define θ(p, x, bond) (resp. θ(p, x, site)) to be the probability that x lies in an infinite open cluster of G in a bond percolation (resp. site percolation) process on G with parameter p. Clearly θ(p, x, bond) and θ(p, x, site) are non-decreasing in p. Also, using the FKG inequality, θ(p, x, bond) ≥ Pp � � {x ↔ y} ∩ {y ↔ ∞} ≥ Pp(x ↔ y)θ(p, y, bond), with a similar inequality for the site process. It follows that the critical points pc(bond) = sup{p : θ(p, x, bond) = 0}, pc(site) = sup{p : θ(p, x, site) = 0}, exist and are independent of the choice of the vertex x. Theorem 5.13. We have that (5.14) 1 ∆ − 1 ≤ pc(bond) ≤ pc(site) ≤ 1 − � 1 − pc(bond) � ∆ . One consequence of this theorem is that pc(bond) < 1 if and only if pc(site) < 1. The third inequality of (5.14) may be improved by replacing the exponent ∆ by ∆ −1, but we do no prove this here. Also, the methods of Chapter 4 may be used to establish the strict inequality pc(bond) < pc(site). See [168] for proofs of the latter facts. Proof. The first inequality of (5.14) follows by counting paths, as in the proof of (3.4). We turn to the remaining two inequalities. Let 0 be a vertex of G, called the origin. We claim that (5.15) C ′ (p, 0, site) ≤ C(p, 0, bond) and (5.16) C(p, 0, bond) ≤ C ′ (p ′ , 0, site) if p ′ ≥ 1 − (1 − p) ∆ , where ≤’ denotes stochastic ordering, and C(p, 0, bond) (resp. C ′ (p, 0, site)) has the law of the cluster of bond percolation at the origin (resp. the cluster of site percolation at the origin conditional on 0 being an open site). Since θ(p, 0, bond) = Prob � |C(p, 0, bond)| = ∞ � , p −1 θ(p, 0, site) = Prob � |C ′ (p, 0, site)| = ∞ � , the remaining claims of (5.14) follow from (5.15)–(5.16).
Percolation and Disordered Systems 175 We construct appropriate couplings in order to prove (5.15)–(5.16). Let ω ∈ {0, 1} E be a realisation of a bond percolation process on G = (V, E) with density p. We may build the cluster at the origin in the following standard manner. Let e1, e2, . . . be a fixed ordering of E. At each stage k of the inductive construction, we shall have a pair (Ak, Bk) where Ak ⊆ V , Bk ⊆ E. Initially we set A0 = {0}, B0 = ∅. Having found (Ak, Bk) for some k, we define (Ak+1, Bk+1) as follows. We find the earliest edge ei in the ordering of E with the following properties: ei /∈ Bk, and ei is incident with exactly one vertex of Ak, say the vertex x. We now set � Ak if ei is closed, (5.17) Ak+1 = Ak ∪ {y} if ei is open, � Bk ∪ {ei} if ei is closed, (5.18) Bk+1 = if ei is open, Bk where ei = 〈x, y〉. If no such edge ei exists, we declare (Ak+1, Bk+1) = (Ak, Bk). The sets Ak, Bk are non-decreasing, and the open cluster at the origin is given by A∞ = limk→∞ Ak. We now augment the above construction in the following way. We colour the vertex 0 red. Furthermore, on obtaining the edge ei given above, we colour the vertex y red if ei is open, and black otherwise. We specify that each vertex is coloured at most once in the construction, in the sense that any vertex y which is obtained at two or more stages is coloured in perpetuity according to the first colour it receives. Let A∞(red) be the set of points connected to the origin by red paths of G. It may be seen that A∞(red) ⊆ A∞, and that A∞(red) has the same distribution as C ′ (p, 0, site). Inequality (5.15) follows. The derivation of (5.16) is similar but slightly more complicated. We start with a directed version of G, namely −→ G = (V, −→ E ) obtained from G by replacing each edge e = 〈x, y〉 by two directed edges, one in each direction, and denoted respectively by [x, y〉 and [y, x〉. We now let −→ ω ∈ {0, 1} −→ E be a realisation of an (oriented) bond percolation process on −→ G with density p. We colour the origin green. We colour a vertex x (�= 0) green if at least one edge f of the form [y, x〉 satisfies −→ ω (f) = 1; otherwise we colour x black. Then (5.19) Pp(x is green) = 1 − (1 − p) ρ(x) ≤ 1 − (1 − p) ∆ , where ρ(x) is the degree of x, and ∆ = maxx ρ(x). We now build a copy A∞ of C(p, 0, bond) more or less as described above in (5.17)–(5.18). The only difference is that, on obtaining the edge ei = 〈x, y〉 where x ∈ Ak, y /∈ Ak, we declare ei to be open for the purpose of (5.17)–(5.18) if and only if −→ ω � [x, y〉 � = 1. Finally, we set A∞(green) to be the set of points connected to the origin by green paths. It may be seen that A∞(green) ⊇ A∞. Furthermore, by (5.19), A∞(green) is no larger in distribution that C ′ (p ′ , 0, site) where p ′ = 1 − (1 − p) ∆ . Inequality (5.16) follows. �
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Page 18 and 19: 158 Geoffrey Grimmett Kesten [201]
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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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