Concentration estimates for the isoperimetric constant of the super critical percolation cluster

We consider the Cheeger constant $\phi(n)$ of the giant component of supercritical bond percolation on $\mathbb{Z}^d/n\mathbb{Z}^d$. We show that the variance of $\phi(n)$ is bounded by $\frac{\xi}{n^d}$, where $\xi$ is a positive constant that depends only on the dimension $d$ and the percolation parameter.


Introduction
Let T d (n) be the d dimensional torus with side length n, i.e, Z d /nZ d , and denote by E d (n) the set of edges of the graph T d (n). Let p c (Z d ) denote the critical value for bond percolation on Z d , and fix some p c (Z d ) < p ≤ 1. We apply a p-bond Bernoulli percolation process on the torus T d (n) and denote by C d (n) the largest open component of the percolated graph (In case of two or more identically sized largest components, choose one by some arbitrary but fixed method). Let Ω = Ω n = {0, 1} E d (n) be the space of configurations for the percolation process and P = P p is the probability measure associated with the percolation process. For a subset A ⊂ C d (n)(ω) we denote by ∂ C d (n) A the boundary of the set A in C d (n), i.e, the set of edges (x, y) ∈ E d (n) such that ω((x, y)) = 1 and with either x ∈ A and y / ∈ A or x / ∈ A and y ∈ A. Throughout this paper c, C and c i denote positive constants which may depend on the dimension d and the percolation parameter p but not on n. The value of the constants may change from one line to the next.
Next we define the Cheeger constant Definition 1.1. For a set ∅ = A ⊂ C d (n) we denote, where | · | denotes the cardinality of a set. The Cheeger constant of C d (n) is defined by: In [BM03] Benjamini and Mossel studied the robustness of the mixing time and Cheeger constant of Z d under a percolation perturbation. They showed that for p c (Z d ) < p < 1 large enough nφ(n) is bounded between two constants with high probability. In [MR04], Mathieu and Remy improved the result and proved the following on the Cheeger constant Theorem 1.2. There exist constants c 2 , c 3 , c > 0 such that for every n ∈ N Recently, Marek Biskup and Gábor Pete brought to our attention that better bounds on the Cheeger constant exist. In [Pet07] and [BBHK08] it is shown that The improved bounds don't improve our result, thus we kept the original [MR04] bounds in our proofs.
Even though the last conjecture is still open, and the expectation of the Cheeger constant is quite evasive, we managed to give a good bound on the variance of the Cheeger constant. This is given in the main Theorem of this paper: Theorem 1.5. There exists a constant ξ = ξ(p, d) > 0 such that A major ingredient of the proof is Talagrand's inequality for concentration of measure on product spaces. This inequality is used by Benjamini, Kalai and Schramm in [BKS03] to prove concentration of first passage percolation distance. A related study that uses another inequality by Talagrand is [AKV02], where Alon, Krivelevich and Vu prove a concentration result for eigenvalues of random symmetric matrices.
Definition 2.2. For n ∈ N we define the following events:

1)
and We start with the following deterministic claim: In order to prove Claim 2.3 we will need the following two lemmas: Lemma 2.4. Fix a configuration ω ∈ Ω and an edge e ∈ E d (n). Let A ⊂ C d (n)(ω e ) be a subset such that |A| = αn d . Then Proof. Since A is a subset of C d (n)(ω e ) it follows that the size of A doesn't change between the configurationsω e andω e and the size of ∂ C d (n) A is changed by at most 1. It therefore follows that Lemma 2.5. Let G be a finite graph, and let A, B ⊂ G be disjoint such that there exists a unique edge e = (x, y), such that x ∈ A and y ∈ B, then Proof. From the assumptions on A and B it follows that and so the lemma follows.
Proof of Claim 2.3. We separate the proof into six different cases according to the following table: • Cases 1 and 2: In those cases the set C d (n) and the edges available from it is the same for both configurations ω and ω e . It therefore follows that ∇ e φ(ω) = 0. See Figure 2.1a, and 2.1b.
• Case 3: In this case the set C d (n) is the same for both configurations ω and ω e , however the set of edges available from C d (n) is increased by one when moving to the configuration ω e , see figure 2.1c. Fix a set A ⊂ C d (n)(ω) of size bigger than c 4 n d which realize the Cheeger constant. It follows that and therefore by Lemma 2.4 we have as required.
• Case 4: We separate this case into two subcases according to the fact weather is an empty set or not. If C d (n)(ω)\C d (n)(ω e ) = ∅ then we are in the same situation as in Case 3, see Figure 2.1d, and so the same argument gives the desired result. So, let us (2.5) Since ω ∈ H 4 n there exists a set A ⊂ C d (n)(ω) of size bigger than c 4 n d realizing the Cheeger constant in the configuration ω. We denote A 1 = A ∩ C d (n)(ω e ) and A 2 = A ∩ (C d (n)(ω)\C d (n)(ω e )). Applying Lemma 2.5 to A 1 and A 2 we see that From (2.5) it follows that |A 2 | ≤ √ n and therefore ψ A 2 (ω) ≥ 1 √ n which gives us that min{ψ A 1 (ω), ψ A 2 (ω)} = ψ A 1 (ω). Indeed, if the last equality doesn't hold then which for large enough n yields a contradiction. Consequently from (2.6) we get that For the other direction, since ω ∈ H 5 n there exists a set B ⊂ C d (n)(ω e ) of size bigger than c 5 n d realizing the Cheeger constant in ω e , then as required.
• Case 5: This case is similar to Case 4, see Figure 2.1f. The proof of this case follows the proof of case 4 above.
• Case 6: This case is impossible by the definition of the set C d (n)(ω).
Next we turn to estimate the probability of the event H n .
Claim 2.6. There exist constants c 1 , c 2 , c 3 , c 4 , c 5 > 0 and a constant c > 0 such that for large enough n ∈ N we have , it's enough to bound each of the last probabilities. The proof of the exponential decay of P((H 1 n ) c ) for appropriate constant is presented in the Appendix. By [MR04] Theorem 3.1 and section 3.4, there exists a c > 0 such that for n large enough, P((H 2 n ) c ) ≤ e −c log 3/2 n for some constants c 2 , c 3 > 0.
Turning to bound P((H 3 n ) c ), we notice that the set C d (n)(ω)△C d (n)(ω e ) is independent of the status of the edge e and therefore (2.8) We already gave appropriate bound for the last term and therefore we are left to bound the probability of {ω ∈ Ω : Notice that the occurrence of this event implies the existence of an open cluster of size bigger than √ n which is not connected to C d (n), and therefore its probability is bounded by is an open cluster that is not connected to C d (n)} ∩ H 1 n ).
By [MR04] Appendix B and [Gri99] Theorem 8.61, for large enough n we have that, is an open cluster that is not connected to the infinite cluster . (2.9) However the probability of the last event decays exponentially with n by [Gri99] Theorem 8.18. In order to deal with the event (H 4 n ) c we define one last event where ǫ(n) = d + 2d log log n log n and (2.10) By [MR04] there exists a constant c > 0 such that for large enough n ∈ N P(G c n ) < e −c log 3 2 n . As before we write and by the probability bound mentioned so far it's enough to bound the probability of the first event (H 4 n ) c ∩H 1 n ∩H 2 n ∩G n . What we will actually show is that for appropriate choice of 0 < c 4 < 1 2 we have (H 4 n ) c ∩ H 1 n ∩ H 2 n ∩ G n = ∅. Indeed, since we assumed the event G n occurs we have that for large enough n ∈ N and every set A ⊂ C d (n)(ω) of size smaller than c 4 n d

It follows that
n . (2.11) Choosing c 4 > 0 such that for large enough n ∈ N we have c 6 c 1/ǫ(n) 4 > c 3 , we get a contradiction to the event H 2 n , which proves that the event is indeed empty. Finally we turn to deal with the event (H 5 n ) c . As before it's enough to bound the probability of the event (H 5 n ) c ∩ H 1 n ∩ H 2 n ∩ H 3 n ∩ H 4 n ∩ G n . We divide the last event into two disjoint events according to the status of the edge e, namely (2.12) and will show that for right choice of c 5 > 0 both V 0 n and V 1 n are empty events. Let us start with V 0 n . Going back to the proof of Claim 2.3 one can see that under the event and therefore φ(ω e ) ≤c 3 n for anyc 3 > c 3 and n ∈ N large enough. (2.14) and therefore A cannot realize the Cheeger constant. On the other hand, if A ⊂ C d (n)(ω e ) satisfy ñ c 3 ≤ |A| ≤ c 5 n d then and therefore (Since we assumed the event G occurs) (2.15) Taking c 5 > 0 small enough such that c 6 2c 1/ǫ(n) 5 − 2c 3 >c 3 we get a contradiction to (2.13). It follows that no set A ⊂ C d (n)(ω e ) of size smaller than c 5 n d can realize the Cheeger constant which contradicts (H 5 n ) c , i.e, V 0 n = ∅. Finally, for V 1 n . The case A ⊂ C d (n)(ω e ) such that |A| < ñ c 3 is the same as for the event V 0 n . If A ⊂ C d (n)(ω e ) satisfy ñ c 3 ≤ |A| ≤ c 5 n d then 8 and therefore as in the case of V 0 n . (2.16) Choosing c 5 small enough, we again get a contradiction to (2.13). and as before this yields that V 1 n = ∅. Proof of theorem 1.5. By [Tal94] (Theorem 1.5) the following inequality holds for some K = K(p), . (2.17) Consequently, if we fix some edge e 0 ∈ E d (n), where the first equality follows from the symmetry of T d (n).

Appendix
In this Appendix for completeness and future reference we sketch a proof of the exponential decay of P((H 1 n ) c ). The proof follows directly from two papers [DP96] by Deuschel and Pistorza and [AP96] by Antal Pisztora, which together gives a proof by a renormalization argument. We borrow the terminology of [AP96] without giving here the definitions.
Lemma 3.1. Let p > p c (Z d ). There exists a c 1 , c > 0 such that for n large enough P p (|C d (n)(ω)| < c 1 n d ) < e −cn .