INEQUALITY OF TWO CRITICAL PROBABILITIES FOR PERCOLATION

For a locally ﬂnite, connected graph G with distinguished vertex 0, let p c ( G ) be the usual critical probability for bond percolation on G , and where ƒ ranges over cutsets (sets of vertices \separating 0 from 1 "), E p refers to (Bernoulli bond) percolation with p the probability that an edge is open, and C (0) is the open cluster containing 0.(The deﬂnition is easily seen to be independent of the choice of distinguished vertex.) We disprove a conjecture of Russ Lyons stating that p cut ( G ) = p c ( G ) for every G , and propose a possible alternative.


Introduction
We consider the usual (Bernoulli) bond percolation on a locally finite, connected graph G = (V, E): each e ∈ E is "open" with probability p, these choices made independently. We write H for the resulting random subgraph of G and Pr p for the law of H. (For percolation background, see e.g. [2] or [5].) For X, Y ⊆ V we denote by {X ←→ Y } the event that some x ∈ X and y ∈ Y are in the same (For connected G this is easily seen to coincide with sup{p : Pr p (|C(x)| = ∞) = 0} for any x ∈ V .) Our focus here is on a second critical probability, introduced by Russ Lyons in [3].
Given 0 ∈ V , say Π ⊆ V is a cutset for 0 if the component of 0 in G − Π is finite ("Π separates 0 from infinity"), and set Again this is easily seen not to depend on our choice of the distinguished vertex. Here we will keep 0 fixed throughout, and Π will always range over cutsets for 0. Note p cut ≤ p c is trivial because of the "first moment method" bound Lyons proved p cut = p c for trees [3] and for certain "tree-like" graphs [4], and suggested ( [3], [4]) the possibility that p cut and p c coincide for every G. That equality also holds for the Euclidean cubic lattices follows from [1], [6].
Here we give a simple family of graphs for which equality does not hold, and, in Section 3, suggest a possible salvage.

Example
A graph G is said to be spherically symmetric about 0 ∈ V if (Aut(G)) 0 is transitive on for each n. For such G, dealing with p cut is simplified by the following observation.

Lemma 2.1 If G is spherically symmetric then (for any p)
inf Proof. Fix a cutset Π, say with Π ⊆ Π 0 ∪ · · · ∪ Π m (of course it is enough to consider minimal Π's, which are necessarily finite). Let P be chosen uniformly from the set of (0, Π m )-paths of length m. By spherical symmetry and

Construction
Set n 0 = 1 and let n 1 , n 2 , . . . be integers with n i ≥ 5n i−1 for i ≥ 1. Set m i = n 1 + · · · + n i . The m i th levels play a special role, and we abbreviate Π mi = Σ i . The structure of G is as follows.
For each i, Σ i is a clique of size 2 ni , and G[∪{Π n : m i−1 ≤ n ≤ m i }] consists of the cliques at levels m i−1 and m i together with, for each v ∈ Σ i−1 , 2 ni−ni−1 paths of length n i joining v to Σ i , the resulting 2 ni paths being disjoint except where they share an end in Σ i−1 .

Analysis
It is easy to see that p c (G) ≥ 1/2: which for p < 1/2 tends to zero as i → ∞.
(The bound is by no means best possible, but there seems no point in trying to optimize either the construction or the analysis, and we will always opt for crude but convenient estimates in what follows.) According to Lemma 2.1, (2) can be established by showing that for and any n, Proof. Fix ε, p as in (3), and write Pr and E for Pr p and E p . For B i as above, Let C i = {H[Σ i ] is connected} and ξ i = 1 − Pr(C i ). A routine calculation gives (say) and the B j 's and C j 's are independent, (5) gives The first expression in brackets is of course the probability that the "direct" (i.e. unique shortest) path from Σ i−1 to v is open. The second is a lower bound on Pr Then (for instance) β n > (1/2)(1 − ε) 4ni/3 2 −ni/2 (since Pr(D i−1 ) > (1 − ε) ni/3 and the maximum in (7) is at least (p ni Pr(B i )(1 − ξ i )) 1/2 > (1/2)(1 − ε) ni 2 −ni/2 ), yielding, finally, E|C(0) ∩ Π n | > |Π n |β n = 2 ni β n > (1/2)(1 − ε) 4ni/3 2 ni/2 > 1.
Remark. As Russ Lyons points out, our construction can easily be modified to have bounded degree; roughly: replace the cliques by expanders and, for each v ∈ Σ i−1 , replace the paths from v to Σ i by a tree of the same depth and with the same leaves, having large (but constant) degrees near the root and paths following this initial stage. We omit the details.

A possible substitute
Following [5], we regard equality of p cut (G) and p c (G) as saying that the first moment bound (1) correctly identifies the critical probability for bond percolation on G. The construction of Section 2 violates this in the natural way, by arranging that the events {0 ←→ v}, v in some cutset Π, are heavily dependent, so that |C(0) ∩ Π| = v∈Π 1 {0←→v} , while usually zero, has a large expectation. This is achieved by making connections within Π easy, so that once we do reach at least one v ∈ Π (an unlikely event), we then typically reach the rest of Π via v. So a natural way to try to salvage the conjecture is to disallow the effects of such "internal" connections, as follows.