Etapas de los planes de carreras

In this section we extend Theorem 10.15 to site percolation. The proof is essen- tially the same, all we have to do is to adapt the probability (1−p)n_appearing

in (31), but we will also adapt Lemma 10.8 in order to obtain a better bound onp_C.

For aP-interface (IV, IO) ofGwe letVO denote the set of vertices incident

with an edge in IO, and we let VV denote the set of vertices incident with an

edge in IV but with no edge in IO. We say that a P-interface I = (IV, IO)

is a site-P-interface, if no edge in IV has both its end-vertices in VO. Note

that any site percolation instanceω ∈ {0,1}V(G)_{naturally gives rise to a bond} percolation instance ω0_{∈ {0,1}}E(G)_{, by settingω}0_{(xy) = 1 wheneverω(x) = 1}

and ω(y) = 1. It is obvious from the definitions that ifI occurs in such an ω0_,

then I is a site-P-interface. For site-P-interfaces we can improve Lemma 10.8 as follows, using the same proof except that we work withGrather thanL(G). Thevertex-boundary size of (IV, IO) is|VV|.

Lemma 10.14. The number of site-P-interfaces(IV, IO)ofGof vertex-boundary

size n such that IV contains a fixed edge of G is less than c0γ˙_Pn, where γ˙P =

(dmP _{−1)e, andd is the degree ofG.}

Using this we can now adapt Theorem 10.12 to site percolation, repeating the proof verbatim, except that we use site-P-interfaces instead ofP-interfaces. Corollary 10.15. LetGbe an 1-ended Cayley graph with a finite presentation

P. Thenp_C≤1−1/γ˙P for site percolation onG.

This bound onp_Cis far from the conjectured p_C=pc, but not so far from

p_C≤1−pc, which is the best that our methods can achieve (and possibly the

truth) in light of a result of Kesten & Zhang, saying that for site percolation on_Zd_{, d≥3, the distribution of the vertex-boundary size of the site-P-interface}

of the cluster of the origin does not have an exponential tail [39, Theorem 4] (here_Zd_{denotes the cubic lattice in}

Rd, and the basisP consists of the squares

bounding the faces of its cubes). Our next result implies that this ‘theoretical barrier’ p_C ≤ 1−pc can in fact be achieved if we are allowed to modify the

graph a little by adding some diagonal edges.

Theorem 10.16. Let Gbe an 1-ended quasi-transitive graph admitting a basis

P of C(G)all cycles of which are triangles. Then p_C≤1−pc for both site and

In particular, we can obtain such a Gby adding to _Zd _{the ‘monotone’ di-}

agonal edges, i.e. the edges of the form xy where yi−xi = 1 for exactly two

coordinatesi≤d, andyi=xi for all other coordinates. Then each square gives

rise to two triangles, and we can use all these triangles as our basis P of the cycle space.

Note that ford= 2 we obtain the triangular lattice, and so Theorem 10.16 can be thought of as a generalisation of Corollary 7.9.

For its proof we will need the following lemma, which is a special case of [51, THEOREM 5.1], the main idea of which we used in Proposition 10.5, as illustrated in Figure 5.

Lemma 10.17. Let G be an 1-ended quasi-transitive graph admitting a basis

P of C(G) all cycles of which are triangles. Then for every site-P-interface (IV, IO)of G, the vertex boundaryVV spans a connected subgraph ofG.

Proof of Theorem 10.16. We first prove the statement for site percolation. We follow the lines of the proof of Theorem 7.1, except that we now letMSndenote

the set of multi-P-interfaces all elements of which are site-P-interfaces. Instead of Lemma 7.6, which states that the boundary of aP-interface spans a connected subgraph of the dual lattice in that setup, we now use Lemma 10.17, which is the analogous statement for the boundary VV of a site-P-interface under our

assumption on P that all its cycles are triangles. The proof of Lemma 7.7 can be repeated verbatim, except that we replace the quasi-geodesic X used there with an arbitrary 2-way infinite quasi geodesic of G, which exists by a standard compactness argument. In that proof, we used the canonical coupling between bond percolation on a planar lattice and its dual, and applied the Aizenman-Barsky property to the subcritical clusters of the dual. Here, we instead use the canonical coupling between site percolation with parameter p

and with parameter 1−pobtained by switching between vacant and occupied vertices. We apply the Aizenman-Barsky property to the boundariesVV of our

site-P-interfaces: since they span connected subgraphs of Gby Lemma 10.17, each suchVV is contained in a cluster of vacant sites. But asp >1−pc, vacant

clusters are subcritical due to that coupling, hence their size distribution has an exponential tail by the Aizenman-Barsky property (Theorem 3.1). The rest of the proof can be repeated as is.

To prove the statement for bond percolation, we use the canonical coupling between bond percolation on G and site percolation on its line graph L(G), noting that the cluster of an edge ofGis infinite in the former if and only if the cluster of the corresponding vertex ofL(G) is infinite in the latter. Our plan is to apply the statement for site percolation we just proved toL(G). Note that if

Gis quasi-transitive, then so is L(G). Moreover, it is straightforward to check that we can obtain a basis ofC(L(G)) from any basisP ofGby adding all the triangles of the formx, y, zinL(G) whenever the edgesx, y, zofGare incident with a common vertex. Thus we can reduce to the case of site percolation as desired.

For both site and bond percolation, sincepc≤pCunlesspc= 1 becauseθ(p)