Vanishing of the anchored isoperimetric profile in bond percolation at p c

We consider the anchored isoperimetric profile of the infinite open cluster, defined for $p>p\_c$, whose existence has been recently proved in [3]. We extend adequately the definition for $p = p\_c$, in finite boxes. We prove a partial result which implies that, if the limit defining the anchored isoperimetric profile at $p\_c$ exists, it has to vanish.


Introduction
The most well-known open question in percolation theory is to prove that the percolation probability vanishes at p c in dimension three.In fact, the interesting quantities associated to the model are very difficult to study at the critical point or in its vicinity.We study here a very modest intermediate question.We consider the anchored isoperimetric profile of the infinite open cluster, defined for p > p c , whose existence has been recently proved in [3].We extend adequately the definition for p = p c , in finite boxes.We prove a partial result which implies that, if the limit defining the anchored isoperimetric profile at p c exists, it has to vanish.The Cheeger constant.For a graph G with vertex set V and edge set E, we define the edge boundary ∂ G A of a subset A of V as We denote by |B| the cardinal of the finite set B. The Cheeger constant of the graph G is defined as This constant was introduced by Cheeger in his thesis [2] in order to obtain a lower bound for the smallest eigenvalue of the Laplacian.
The anchored isoperimetric profile ϕ n (p).Let d ≥ 2. We consider an i.i.d.supercritical bond percolation on Z d , every edge is open with a probability p > p c (d), where p c (d) denotes the critical parameter for this percolation.We know that there exists almost surely a unique infinite open cluster C ∞ [5].We say that H is a valid subgraph of C ∞ if H is connected and 0 ∈ H ⊂ C ∞ .We define the anchored isoperimetric profile ϕ n (p) of C ∞ as follows.We condition on the event {0 ∈ C ∞ } and we set The following theorem from [3] asserts the existence of the limit of nϕ n (p) when p > p c (d).We wish to study how this limit behaves when p is getting closer to p c .To do so, we need to extend the definition of the anchored isoperimetric profile so that it is well defined at p c (d).We say that H is a valid subgraph of C(0), the open cluster of 0, if H is connected and 0 ∈ H ⊂ C(0).We define ϕ n (p) for every p ∈ [0, 1] as In where θ(p) is the probability that 0 belongs to an infinite open cluster.The techniques of [3] to prove the existence of this limit rely on coarse-graining estimates which can be employed only in the supercritical regime.Therefore we are not able so far to extend the above convergence at the critical point p c .Naturally, we expect that n ϕ n (p c ) converges towards 0 as n goes to infinity, unfortunately we are only able to prove a weaker statement.
Theorem 1.2.With probability one, we have We shall prove this theorem by contradiction.We first define an exploration process of the cluster of 0 that remains inside the box [−n, n] d .If the statement of the theorem does not hold, then the cluster of 0 satisfies a d-dimensional anchored isoperimetric inequality.It follows that the number of sites that are revealed in the exploration of the cluster of 0 will grow fast enough of order n d−1 .Then, we can prove that the intersection of the cluster that we have explored with the boundary of the box [−n, n] d is of order n d−1 .Using the fact that there is no percolation in a half-space, we obtain a contradiction.Before starting the precise proof, we recall some results from [3] on the meaning of the limiting value ϕ(p).
We consider the anisotropic isoperimetric problem associated with the norm τ : The famous Wulff construction provides a minimizer for this anisotropic isoperimetric problem.We define the set W τ as where • denotes the standard scalar product and S d−1 is the unit sphere of R d .Up to translation and Lebesgue negligible sets, the set is the unique solution to the problem (1).

Representation of ϕ(p).
In [3], we build an appropriate norm β p for our problem that is directly related to the open edge boundary.We define the Wulff crystal W p as the dilate of W βp such that L d (W p ) = 1/θ(p), where θ(p) = P(0 ∈ C ∞ ).We denote by I p the surface tension associated with the norm β p .In [3], we prove that

Proofs
We prove next the following lemma, which is based on two important results due to Zhang [9] and Rossignol and Théret [6].To alleviate the notation, the critical point p c (d) is denoted simply by p c .Let B be a subset of R d having a regular boundary and such that L d (B) = 1/δ.As the map p → θ(p) is non-decreasing and L d (W p ) = 1/θ(p), we have Moreover as W p is the dilate of the minimizer associated to the isoperimetric problem (1), we have In [9], Zhang proved that β pc = 0.In [6], Rossignol and Théret proved the continuity of the flow constant.Combining these two results, we get that lim p→pc p>pc β p = β pc = 0 and so lim p→pc p>pc Finally, we obtain lim p→pc p>pc This yields the result.
Proof of theorem 1.2.We assume by contradiction that Therefore there exist positive constants c and δ such that Therefore, there exists a positive integer n 0 such that In what follows, we condition on the event Note that on this event, 0 is connected to infinity by a p c -open path.For H a subgraph of Z d , we define We define next an exploration process of the cluster of 0. We set C 0 = {0}, A 0 = ∅.Let us assume that C 0 , . . ., C l and A 0 , . . ., A l are already constructed.We define We have Since A l+1 and C l are disjoint, we have Let us set α = 1/n d 0 so that |C 0 | = αn d 0 .Let k be the smallest integer greater than 2 d+1 d/c.We recall that c and n 0 were defined in ( 2) and ( 3).Let us prove by induction on n that This is true for n = n 0 .Let us assume that this inequality is true for some integer In this case, for any integer l ≤ k, we have also As k ≥ 2 d+1 d/c, we get This concludes the induction.Let η > 0 be a constant that we will choose later.In [1], Barsky, Grimmett and Newman proved that there is no percolation in a half-space at criticality.An important consequence of the result of Grimmett and Marstrand [4] is that the critical value for bond percolation in a half-space equals to the critical parameter p c (d) of bond percolation in the whole space, i.e., we have P(0 is connected to infinity by a p c -open path in N × Z d−1 ) = 0 , so that for n large enough, In what follows, we will consider an integer n such that the above inequality holds.By construction the set C n is inside the box [−n, n] d .Starting from this cluster, we are going to resume our exploration but with the constraint that we do not explore anything outside the box [−n, n] d .We set C ′ 0 = C n and A ′ 0 = ∅.Let us assume C ′ 0 , . . ., C ′ l and A ′ 0 , . . ., A ′ l are already constructed.We define We stop the process when A ′ l+1 = ∅.As the number of vertices in the box [−n, n] d is finite, this process of exploration will eventually stop for some integer l.We have that |C ′ l | ≤ n d and n φk (p c ) > c so that Moreover, for n ≥ kn 0 , we have, thanks to inequality (5), We suppose that n is large enough so that n ≥ kn 0 and ⌊ n k ⌋ ≥ n/2k.Combining the two previous display inequalities, we conclude that Therefore, for n large enough, there exists one face of [−n, n] d such that there are at least cαn Moreover, we have Finally, combining inequalities ( 7) and (8), we get Therefore, we can choose η small enough such that .

The
Wulff theorem.We denote by L d the d-dimensional Lebesgue measure and by H d−1 denotes the (d − 1)-Hausdorff measure in dimension d.Given a norm τ on R d and a subset E of R d having a regular enough boundary, we define I τ (E), the surface tension of E for the norm τ , as

≥ δ 2 . ( 6 )
d−1 /(2 d k d 2d) vertices that are connected to 0 by a p c -open path that remains inside the box [−n, n] d and so P   there exists one face of [−n, n] d with at least cαn d−1 /(2 d k d 2d) vertices that are connected to 0 by a p c -open path that remains inside the box [−n, n] d   Let us denote by X n the number of vertices in the face {−n} × [−n, n] d−1 that are connected to 0 by a p c -open path inside the box [−n, n] d .We have E(X n ) ≤ ({−n} × [−n, n] d−1 ) ∩ Z d P   ∃γ a p c -open path starting from 0 in N × Z d−1 such that |γ| ≥ n   ≤ (2n + 1) d−1 η .
and so using the symmetry of the lattice P   there exists one face of [−n, n] d such there are at least cαn d−1 /(2 d k d 2d) vertices that are connected to 0 by a p c -open path that remains inside the box [−n, n] d   ≤ 2d P X n > cα 2d2 d k d n d−1 ≤ δ 5