A lower bound for point-to-point connection probabilities in critical percolation

Consider critical site percolation on $\mathbb{Z}^d$ with $d \geq 2$. We prove a lower bound of order $n^{- d^2}$ for point-to-point connection probabilities, where $n$ is the distance between the points. Most of the work in our proof concerns a `construction' which finally reduces the problem to a topological one. This is then solved by applying a topological fact, which follows from Brouwer's fixed point theorem. Our bound improves the lower bound with exponent $2 d (d-1)$, used by Cerf in 2015 to obtain an upper bound for the so-called two-arm probabilities. Apart from being of interest in itself, our result gives a small improvement of the bound on the two-arm exponent found by Cerf.


Introduction and statement of the main results
Consider site percolation on Z d with d ≥ 2. Let p c = p c (d) denote the critical probability. We use the notation Λ(n) for the box [−n, n] d . Our main result is the following, Theorem 1.1 below. Although the result is, essentially, only relevant for dimensions 3 -6, we formulate it for all d ≥ 2 because the proof works for all these values of d 'simultaneously'.
There is a constant c > 0, which depends only on the dimension d, such that for all x ∈ Z d , x = 0, Here x := max 1≤i≤ |x i |, and {0 ↔ x in Λ(4 x )} is the event that there is an open path from 0 to x which lies in Λ(4 x ).
Let us first explain why this result is, essentially, only relevant for dimensions 3 -6: For 'nice' two-dimensional lattices (including the square lattice) a much better exponent (2/3 or even smaller) than the exponent 4 from (1) is known. For these lattices it is believed (and known for site percolation on the triangular lattice from [7]) that for every C > 1, P pc (0 ↔ x in Λ(C x )) ≈ x −5/24 . Further, it has been proved for dimensions d ≥ 11 and is strongly believed for d ≥ 7, that the 'unrestricted' event in (1) (i.e. without the restriction "in Λ(4 x )") has probability of order x −(d−2) , see [5] and [6]. Moreover, it seems to be believed that for such d a suitably restricted form of this event (with a suitable constant instead of the factor 4 in the l.h.s. of (1)) also has probability of order x −(d−2) .
The methods used for the above mentioned results for d = 2 don't work for d ∈ {3, . . . , 6}, and presumably this also holds for the methods used for d ≥ 11. To our knowledge, for dimension d ∈ {3, . . . , 6} no version of Theorem 1.1 with exponent ≤ d 2 exists in the literature, even if we drop the restriction "in Λ(4 x )". In fact, we don't know if the probabilities for the events with and without the restriction are really of different order. We stated our result for the case with restriction because it is stronger, and because of its applicability for other purposes, see the Remarks below. Theorem 1.1 has the following implication: There is a constant c ′ > 0, which depends only on the dimension d, such that for all n ≥ 1, and all x, y ∈ Λ(n), Indeed, this follows readily from Theorem 1.1 by observing that, for all x, y ∈ Λ(n), we have that y − x ≤ 2n and that P pc (x ↔ y in Λ(9n)) ≥ P pc (x ↔ y in x + Λ(8n)), which by translation invariance is equal to P pc (0 ↔ y − x in Λ(8n)).
Corollary 1.2 clearly improves (apart from the factor 9 in the l.h.s. of (2), see Remark (ii) below) the following result by Cerf, which is one of the motivations for our work: There is a constantc > 0, which depends only on the dimension d, such that for all n ≥ 1, and all x, y ∈ Λ(n), Remarks: (i) Lemma 1.3 above was used in a clever way by Cerf in [1] to obtain new upper bounds for two-arm probabililities. Cerf's work on two-arm probabililities is, in turn, mainly motivated by the famous conjecture that, for any dimension ≥ 2, there is no infinite open cluster at p c . This conjecture, one of the main open problems in percolation theory, has only been proved for dimension two and for high dimensions.
(ii) The precise factor 2 in the expression Λ(2n) in (3) is not essential for Cerf's applications mentioned in Remark (i): a bigger constant, which may even depend on the dimension, would work as well (with tiny, straightforward modifications of Cerf's arguments). This is why we have not seriously tried to reduce the factor 9 in the expression Λ(9n) in Corollary 1.2.
(iii) What does matter for Cerf's applications is an improvement of the power 2d(d − 1) in the r.h.s. of (3). By a simple adaptation of a step in Cerf's proof of Lemma 1.3, the exponent 2d(d − 1) can be replaced by (2d − 1)(d − 1). Our Corollary 1.2 states that it can be replaced by d 2 . So, for instance, for d = 3 the exponents given by Cerf, by the small adaptation of Cerf's argument mentioned above, and by our Corollary 1.2, are 12, 10 and 9 respectively; and for d = 4 they are 24, 21 and 16 respectively. Following step by step the computations in Sections 7-9 in Cerf's paper, shows that our improvement of Lemma 1.3 also provides a small improvement of Cerf's lower bound for the two-arm exponent.

Main ideas in the proof of Theorem 1.1
We first present, somewhat informally, Cerf's proof of Lemma 1.3 (and the proof of the small adaptation mentioned in Remark (iii) above).
To focus on the main idea we ignore here the restriction "in Λ(2n)" in Lemma 1.3. The key ingredient is a result, Corollary 2.3 below, which goes back to work by Hammersley. Applied to the special case where Γ is a box centered at 0, and using symmetry, this result gives that there is a C = C(d) such that for every n there is a 'special' site v n on the boundary of Λ(n) with v n 1 = n and such that Now let x be a vertex. By symmetry we may assume that its coordinates x i , i = 1, . . . , d, are non-negative. For simplicity we also assume that they are even. Let x(i) be the vertex of which the ith coordinate is equal to x i and all other coordinates are 0.
We have by translation invariance and FKG that P pc (0 ↔ x) is larger than or equal to Using symmetry, the ith factor in this product is equal to P pc (0 ↔ (x i , 0, 0, · · · , 0)), and hence, by FKG (and again by symmetry), at least . Hence the product of the d factors is at least of order x −2d(d−1) . This gives essentially (apart from some relatively straightforward issues) Lemma 1.3.
The small adaptation mentioned before comes from an observation concerning the first step of the argument above: Note that P pc (0 ↔ x) is also larger than or equal to The first factor is at least C · x −(d−1) . And the second factor can be written as P pc (0 ↔ v x 1 −x) to which we can apply Cerf's argument mentioned above. However, since v x 1 −x has first coordinate 0, and hence has at most d−1 nonzero coordinates, that argument gives a lower bound of order x −2(d−1)(d−1) . Hence, the product of the two factors is at least of order So, roughly speaking, the explanation of Lemma 1.3 is that x can be written as the sum of 2d (and in fact, as the above adaptation showed, 2d−1) special points, where each special point 'costs' a factor of order x −(d−1) .
Our proof of Theorem 1.1 (and thus of Corollary 1.2) also uses the idea of certain special points, which will be called 'good' points. Our notion of good points is weaker than that of the special points mentioned above, in the sense that they 'cost' a factor of order x −d . However, we will prove that, roughly speaking, each point is the sum of 'just' d good points, which together with the previous statement gives the exponent d 2 in Theorem 1.1.
To prove this we first show, again using Corollary 2.3 (but now with general Γ, not only boxes) the existence of suitable paths of good points and then turn the problem into a topological issue. This is then finally solved by applying the 'topological fact' Lemma 2.11. Before we start, we reformulate Theorem 1.1 as follows: There is a constant c > 0, which depends only on the dimension d, such that for all n ≥ 1, and all x ∈ Λ(n), This proposition is trivially equivalent to Theorem 1.1. This reformulation is less 'compact' than that of Theorem 1.1 but has the advantage that it is more natural with respect to the approach in our proof (where, as we will see, we first fix an n and then distinguish between 'good' and other points in Λ(n).

Preliminaries
First we introduce some notation and definitions: For two vertices v, w, the notation v ∼ w means that v and w are neighbours, i.e. that v − w 1 : (Note that we allow consecutive vertices to be equal; this is done for convenience later in this paper.) If W is a set of vertices, we define We will use a result, Lemma 2.2 below, which goes back to work by Hammersley in the late fifties. Hammersley only proved the special case (used by Cerf) where Γ is of the form Λ(k). The proof of the general case (which we use) is very similar, it is given in Section 2 of [2]. [2].) Let Γ be a finite, connected set of vertices containing 0. Then

Lemma 2.2. (Hammersley [4], Duminil-Copin and Tassion
where {0 ↔ x 'in' Γ} is the event that there is an open path from 0 to x of which all vertices, except x itself, are in Γ.
Clearly, this lemma has the following consequence.

The set of 'good' vertices and its properties
Since, throughout the proof, the percolation parameter p is equal to p c , we simply write P instead of P pc . Let n ≥ 1 be fixed, and consider the box Λ(n).

Topological approach
We use the curves c (i) , i = 1, . . . , d, introduced in the previous subsection, to define a (as it turns out, useful) function g : Next, we introduce functions h d : R d → R defined recursively for d ≥ 1 by By induction it easily follows that for all d ≥ 1 and all x ∈ R d there exist weights a j ∈ {−1, 1} for j = 1, . . . , d such that Now we define g : [0, 1] d → R d by describing the d coordinates of g(t).
Lemma 2.9. The function g : [0, 1] d → R d has the following properties: (a) g is continuous; Proof of Lemma 2.9. Part (a) follows because g is a composition of continuous functions. Note that C i,j (t) = (c (j) (t j )) i so that |C i,j (t)| ≤ n for all i, j by Lemma 2.7(ii) and (iii). By (9), this implies that Together with Lemma 2.7(ii), this proves part (b). By definition of g and by (9), for each t ∈ [0, 1] d there exist weights a ij (t) ∈ {−1, 1} such that where 1 denotes (1, . . . , 1) andC(t) is the matrix with elementsC i,j (t) = a ij (t)C i,j (t) and columnsc (j) (t j ). For all i, the partial sums of (10) satisfy n , and by noting that Lemma 2.9 (a) and (b) guarantees that this function satisfies the conditions of Lemma 2.11. We don't know if this lemma is, more or less explicitly, in the literature. We are grateful to Lex Schrijver for providing his proof.
For all i ∈ {1, . . . , d} and all x ∈ [0, 1] d with Now suppose there is an a ∈ (0, 1) d ) which is not in f ([0, 1] d ). We will see that this leads to a contradiction. Fix such an a and let π be the projection from a on ∂