INFORMACIÓN GENERAL

Define theboundary sizeof aP-interfaceI= (IV, IO) to be|IV|. Note that every

to a connected induced subgraph of themth power of the line graph L(G)m _of

G, wherem=mP :=bt/2candt is the length of the longest cycle inP. The

degree of each vertex ofL(G) is at most 2d−2, wheredis the maximum degree ofG(we are still assuming thatGsatisfies (29)), and so the degree of each vertex ofL(G)m_{is at most (2d−2)}m_{. Applying the remark after Proposition 14.1 to}

L(G)m_{, combined with the fact that any P-interface I = (I}

V, IO) is uniquely

determined byIV by the definitions, we thus deduce that

Lemma 10.8. The number of P-interfaces (IV, IO) of G of boundary size n

such that IV contains a fixed edge ofGis less than cγPn, wherec is a constant, γP = ((2d−2)mP −1)e, anddis the maximum degree ofG.

The following is the analogue of Proposition 7.5.

Lemma 10.9. For everyP-interfaceI= (IV, IO)ofG, we have|IV| ≥ |IO|/dt,

where dis the maximum degree ofG andt is the length of the longest cycle in

P.

Proof. By (iv) of Definition 10.3, eache∈IO has distance less than tfrom IV

in the subgraphGI ofGspanned byIV∪IO. Using this fact we can assign each

e∈IO to an edgef(e) of IV so that the distance betweene andf(e) inGI is

less thant. Then the number|f−1_{(g)|of edges ofI}

O assigned to anyg∈IV is

at most the size of the ball of radiust−1 aroundginG, which is at mostdt−1 sinceGisd-regular. Thus |IV| ≥ |IO|/dt−1 by the pigeonhole principle.

Let R = . . . , r−1, r0, r1, . . . be 2-way infinite geodesic with r0 =o (such a geodesic exists in every Cayley graph by an elementary compactness argument, provided we assume e.g. the Axiom of Countable Choice). Let fi denote the

edgeriri+1 ofR.

Lemma 10.10. For every P-interface I = (IV, IO) of o with boundary size

|IV|=n, the setIV contains at least one of the edgesf0, f1, . . . , fdt_n−1.

Proof. Each of the two 1-way infinite subpaths ofRstarting atoconnectsoto infinity, so IV must contain an edge from each of them. By Proposition 10.7

and Proposition 10.9, IO is connected, incident to both of these edges, and

|IO| ≤dtn. Thus ifIV contains some edgeriri+1 withi≥dtn, then it cannot meet. . . r−1r0 becauseR is a geodesic.

A multi-P-interface S is a finite set of P-interfaces {(Ii

V, IOi)}1≤i≤k such

that the corresponding graphs Gi_O, i.e. the subgraphs of G spanned by the edges in I_Oi, are pairwise vertex disjoint. Define theboundary ∂S of S to be

S

1≤i≤k|I i

V|. Let MS denote the set of multi-P-interfaces and MSn the set

of multi-P-interfaces of total boundary size n. Using the above lemma and Proposition 10.10 we can upper bound the number of elements of MSn that

can occur simultaneously in anyω similarly to the proof of Proposition 7.7. Lemma 10.11. There is a constant x∈_Rsuch that for every n∈_Nat most

x√n _{elements ofMS}

n can occur simultaneously in anyω.

We will now prove that p_C < 1 for every finitely presented Cayley graph following the approach of Section 7.2, replacing the use of exponential decay of the dual by Lemmas 10.8 and 10.9.

Theorem 10.12. LetG be an 1-ended Cayley graph with a finite presentation

P. Thenp_C≤1−1/γP for bond percolation on G.

Proof. Similarly to (11), we claim that

1−θo(p) =PS∈MS(−1)c(S)+1QS(p) (30)

for everyp∈(q,1], wherec(S) denotes the number ofP-interfaces in the multi-

P-interfaceS, and QS(p) :=Pp(S occurs).

We will use Proposition 10.4 to prove that the above formula holds. By that proposition, C(o) is finite if and only if it meets a P-interface. Since for any pair of distinct occuringP-interfaces the graphsGO do not share a vertex, the

inclusion-exclusion principle yields

1−θo(p) =P( at least oneP-interface occurs ) =

X

S∈MS

(−1)c(S)+1QS(p)

provided the latter sum converges absolutely. Once again X S∈MSn QS(p) =Ep( X S∈MSn χ{Soccurs})

and by Proposition 10.11 we conclude that

X S∈MSn QS(p)≤x √ n Pp(someS∈ MSn occurs).

The event {some S ∈ MSn occurs} implies that a set of edges with certain

properties is vacant and our goal is to use Peierls’ argument to conclude that the probability of the latter event decays exponentially for large enoughp.

LetS ∈ MSn and letX1, X2, . . . , Xk be the components of the subgraph

of L(G)m _{spanned by ∂S, where m = bt/2c. By the argument at the begin-}

ning of Section 10.5, each Xi contains the boundary of a P-interface of size at

most ni := |Xi|. Thus by Proposition 10.10, Xi contains one of the edges of

f0, f1, . . . , fdt_n

i−1. The Hardy–Ramanujan formula and Proposition 10.8 now

easily yield that the number of all possible boundaries ofMSn is at most

r √

n_max{ck_dkt_n

1n2. . . nk}γPn,

where the maximum ranges over all partitions {n1, n2, . . . , nk} of n such that

every N appears at mostdt_{N times. As in the proof of Theorem 7.1, it is easy}

to check that the quantity max{ck_dkt_n

1n2. . . nk}grows subexponentially inn.

Since eachS∈ MSn occurs with probability at most (1−p)nby the definitions,

we conclude that

Pp(someS∈ MSn occurs)≤r √

n_max{ck_dkt_n

1n2. . . nk}γPn(1−p)n, (31)

and thus Pp(someS ∈ MSn occurs) decays exponentially for every p > 1−

1/γP.

Finally, combining this exponential decay with Proposition 4.14 and Proposition 10.9 we deduce thatθis analytic in (1−1/γP,1], arguing as in the

end of the proof of Theorem 7.1.

Corollary 10.13. ForG =_Zd _{we have p}

C≤1−1/γd where

γd= ((4d−2)2−1)e.

Here, _Zd _{denotes the cubic lattice in}

Rd. That pC <1 forG =Z

d _{is also}

proved in [16] and [15]; our bounds are better than those of [16] and worse than those of [15]. They could be improved if we had a more precise upper bound on the number ofP-interfaces in this case than the one provided by Lemma 10.8. Our proof that p_C <1 can be extended to any quasi-transitive lattice in Rd.

For this we need to show that such a graph admits a basis of its cycle space with bounded cycle lengths, but this is just an exercise.