Expansion in n−1 for percolation critical values on the n-cube and Zn: the first three terms 4 0 0 2 Remco van der Hofstad∗ Gordon Slade† n a December 22, 2003 J 8 ] R Abstract P . h Let p (Q ) and p (Zn) denote the critical values for nearest-neighbour bond percolation c n c t a on the n-cube Q = {0,1}n and on Zn, respectively. Let Ω = n for G = Q and Ω = 2n for n n m G = Zn denote the degree of G. We use the lace expansion to prove that for both G = Q n [ and G = Zn, 1 7 v p (G) = Ω−1+Ω−2+ Ω−3+O(Ω−4). c 2 2 7 0 This extends by two terms the result p (Q ) = Ω−1 + O(Ω−2) of Borgs, Chayes, van der c n 1 Hofstad, Slade and Spencer, and provides a simplified proof of a previous result of Hara and 0 4 Slade for Zn. 0 / h 1 Main result t a m : We consider bond percolation on Zn with edge set consisting of pairs {x,y} of vertices in Zn with v kx−yk = 1, where kwk = n |w | for w ∈ Zn. Bonds (edges) are independently occupied with i 1 1 j=1 j X probability p and vacant with probability 1−p. We also consider bond percolation on the n-cube P r a Q , which has vertex set {0,1}n and edge set consisting of pairs {x,y} of vertices in {0,1}n with n kx−yk = 1, where we regard Q as an additive group with addition component-wise modulo 2. 1 n Again bonds are independently occupied with probability p and vacant with probability 1−p. We write G in place of Q and Zn when we wish to refer to both models simultaneously. We write Ω n for the degree of G, so that Ω = 2n for Zn and Ω = n for Q . n For the case of Zn, the critical value is defined by p (Zn) = inf{p : ∃ an infinite connected cluster of occupied bonds a.s.}. (1.1) c Given a vertex x of G, let C(x) denote the connected cluster of x, i.e., the set of vertices y such that y is connected to x by a path consisting of occupied bonds. Let |C(x)| denote the cardinality ∗Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. [email protected] †Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada. [email protected] 1 of C(x), and let χ(p) = E |C(0)| denote the expected cluster size of the origin. Results of [1, 20] p imply that p (Zn) = sup{p : χ(p) < ∞}. (1.2) c is an equivalent definition of the critical value. For percolation on a finite graph G, such as Q , the above characterizations of p (G) are n c inapplicable. In [8, 9, 10] (in particular, see [10]), it was shown that there is a small positive constant λ such that the critical value p (Q ) = p (Q ;λ ) for the n-cube is defined implicitly by 0 c n c n 0 χ(p (Q )) = λ 2n/3. (1.3) c n 0 Given λ , (1.3) uniquely specifies p (Q ), since χ(p) is a polynomial in p that increases from 0 c n χ(0) = 1 to χ(1) = 2n. Our main result is the following theorem. Theorem 1.1. (i) For G = Zn, 1 1 7 1 1 p (Zn) = + + +O as n → ∞. (1.4) c 2n (2n)2 2(2n)3 (2n)4 (cid:16) (cid:17) (ii) For Q , fix constants c,c′ independent of n, and choose p such that χ(p) ∈ [cn3,c′n−62n] (e.g., n p = p (Q ;λ )). Then c n 0 1 1 7 1 1 p = + + +O as n → ∞. (1.5) n n2 2n3 n4 (cid:16) (cid:17) The constant in the error term depends on c,c′, but does not depend otherwise on p. By Theorem 1.1, the expansions of p (G) in powers of Ω−1 are the same for Q and Zn, up to c n and including order Ω−3. Higher order coefficients could be computed using our methods, but the labour cost increases sharply with each subsequent term. Although we stop short of computing the coefficient of Ω−4, we expect that the coefficients for Q and Zn will differ at this order. In n [18], for both Q and Zn, we prove the existence of asymptotic expansions for p (G) to all orders n c in Ω−1, without computing the numerical values of the coefficients. ForQ ,itwasshownbyAjtai,Koml´osandSzemer´edi[3]thatp (Q ) > n−1(1+ǫ)foreveryfixed n c n ǫ > 0 (although the above definition of p (Q ) did not appear until [8]). Bollob´as, Kohayakawa c n and L uczak [7] improved this to p (Q ) ∈ [1−e−o(n), 1 + 60(logn)3]. Theorem 1.1 extends the very c n n−1 n n2 recent result p (Q ) = n−1+O(n−2) of [8, 9] by two terms. Bollob´as, Kohayakawa and L uczak [7] c n raised the question of whether the critical value might be equal to 1 , but we see from (1.5) that n−1 p (Q ) = 1 + 5n−3 +O(n−4). c n n−1 2 For Zn, Theorem 1.1 is identical to a result of Hara and Slade [16, 17]. Earlier, Bollob´as and Kohayakawa [6], Gordon [13], Kesten [19] and Hara and Slade [15] obtained the first term in (1.4) for Zn with error terms O((logn)2n−2), O(n−65/64), O((loglogn)2(nlogn)−1) and O(n−2), respectively. Recently, Alon, Benjamini and Stacey [4] gave an alternate proof that p (Zn) is c asymptotic to (2n)−1 as n → ∞. The expansion 1 1 7 16 103 p (Zn) = + + + + +··· (1.6) c 2n (2n)2 2(2n)3 (2n)4 (2n)5 2 was reported in [12], but with no rigorous bound on the remainder. We remark that for oriented percolation on Zn, defined in such a way that the forward degree is n, it was proved in [11] that the critical value obeys the bounds 1 1 1 1 1 1 1 + +o ≤ p (oriented Zn) ≤ + +O . (1.7) n 2n3 n3 c n n3 n4 (cid:18) (cid:19) (cid:18) (cid:19) Our method is based on the lace expansion and applies the general approach of [16, 17] that was used to prove Theorem 1.1(i) for Zn, but our method here is simpler and applies to Zn and Q simultaneously. n Remark. For Q , it is a direct consequence of [18, Proposition 1.2] that if there is some sequence n p (depending on n) with χ(p) ∈ [cn3,c′n−62n] such that p = n−1 + n−2 + 7n−3 + O(n−4), then 2 the same asymptotic formula holds for all such p. Thus it suffices to prove (1.5) for a single such sequence p. We fix some sequence f such that lim f n−M = ∞ for every positive integer n n→∞ n M and such that lim f e−αn = 0 for every α > 0. We define p¯ by χ(p¯) = f , and observe n→∞ n n that eventually χ(p¯) ∈ [cn3,c′n−62n]. For G = Q , it therefore suffices to prove that p¯ has the n expansion (1.5). We will use the notation p¯ (G = Q ), p¯ = p¯ (G) = n (1.8) c c p (Zn) (G = Zn).  c  2 Application of the lace expansion For Q or Zn with n large, the lace expansion [15] gives rise to an identity n 1+Πˆ p χ(p) = , (2.1) 1−Ωp[1+Πˆ ] p where Πˆ is a function that is finite for p ≤ p (G). Although we do not display the dependence p c explicitly in the notation, Πˆ does depend on the graph Q or Zn. The identity (2.1) is valid for p n p ≤ p (G). For a derivation of the lace expansion, see, e.g., [9, Section 3]. It follows from (2.1) c that 1 Ωp = −χ(p)−1. (2.2) 1+Πˆ p The function Πˆ has the form p ∞ Πˆ = (−1)NΠˆ(N), (2.3) p p N=0 X with (recall (1.8)) C N∨1 |Πˆ(N)| ≤ uniformly in p ≤ p¯ . (2.4) p Ω c (cid:18) (cid:19) For Q , the formula (2.1) and the bounds (2.4) are given in [9, (6.1)] and [9, Lemma 5.4], re- n spectively (with our Πˆ written as Πˆ (0)). In more detail, [9, Lemma 5.4] states that Πˆ(N) ≤ p p p [const(λ3 ∨ β)]N∨1, where λ = χ(p)2−n/3 ≤ f 2−n/3 for p ≤ p¯ (Q ). By definition, f 2−n/3 is n c n n 3 exponentially small in n. In addition, it is shown in [9, Proposition 2.1] that β can be chosen proportional to n−1. It follows from (2.2) that 1 np¯ (Q ) = +O(f−1). (2.5) c n 1+Πˆ n p¯c(Qn) The second term on the right hand side of (2.5) can be neglected in the proof of Theorem 1.1. Equations (2.3)–(2.5) give p¯ (Q ) = n−1 +O(n−2). c n For Zn, (2.1) and (2.4) follow from results in [15, Section 4.3.2]. (Note the notational difference thatin[15]whatwearecallinghereΠˆ(N) iscalledgˆ (0)andthatΠˆ(N) in[15]issomethingdifferent.) p N p Since χ(p (Zn)) = ∞, it follows from (2.2) that c 1 2np (Zn) = . (2.6) c 1+Πˆ pc(Zn) With (2.3)–(2.4), this implies that p (Zn) = (2n)−1 +O(n−2). c The identities (2.5) and (2.6) give recursive equations for p¯ . To prove Theorem 1.1 using this c recursion, we will apply the following proposition. In its statement, we write n−1 for Q Ω′ = n (2.7) 2n−2 for Zn.  Proposition 2.1. For G = Zn and G = Q, uniformly in p ≤ p¯ (G), n c 3 Πˆ(0) = ΩΩ′p4 +O(Ω−3), (2.8) p 2 Πˆ(1) = Ωp2 +4ΩΩ′p4 +O(Ω−3), (2.9) p Πˆ(2) = Ωp3 +Ω(Ω−1)p4 +O(Ω−3), (2.10) p ∞ Πˆ(N) = O(Ω−3). (2.11) p N=3 X We show now that Proposition 2.1 implies Theorem 1.1. It follows from Ωp¯ (G) = 1+O(Ω−1) c (as noted below (2.5) and (2.6)), (2.3), and Proposition 2.1 that 1 Πˆ = − +O(Ω−2). (2.12) p¯c(G) Ω With (2.5)–(2.6), this implies that 1 Ωp¯ (G) = 1+ +O(Ω−2). (2.13) c Ω Using this in the bounds of Proposition 2.1, along with (2.3), gives 3 1 1 4 1 1 Πˆ = −Ω( + )2 − + + +O(Ω−3) p¯c(G) 2Ω2 Ω Ω2 Ω2 Ω2 Ω2 1 5 = − − +O(Ω−3). (2.14) Ω 2Ω2 4 Substitution of this improvement of (2.12) into (2.5)–(2.6) then gives 1 7 Ωp¯ (G) = 1+ + +O(Ω−3). (2.15) c Ω 2Ω2 Thus, to prove Theorem 1.1, it suffices to prove Proposition 2.1. Since (2.11) is a consequence of (2.4), we must prove (2.8)–(2.10). Precise definitions of Πˆ(N), for N = 0,1,2, will be given in p Section 4. 3 Preliminaries Before proving Proposition 2.1, we recall and extend some estimates from [9, 15]. Let D(x) = Ω−1 if x is adjacent to 0, and D(x) = 0 otherwise. Thus D(y−x) is the transition probability for simple random walk on G to make a step from x to y. Let τ (y −x) = P (x ↔ y) p p denote the two-point function. For i ≥ 0, we denote by {x ←−→ y} (3.1) i the event that x is connected to y by an occupied (self-avoiding) path of length at least i, and define τ(i)(x,y) = P(x ←−→ y). (3.2) p i We define the Fourier transform of an absolutely summable function f on the vertex set V of G by fˆ(k) = f(x)eik·x (k ∈ V∗), (3.3) x∈V X where V∗ = {0,π}n for Q and V∗ = [−π,π]n for Zn. We write the inverse Fourier transform as n f(x) = fˆ(k)e−ik·x, (3.4) Z where we use the convenient notation 2−n gˆ(k) (G = Q ) k∈{0,π}n n gˆ(k) = (3.5)  gˆ(k) dnk (G = Zn). Z  [−π,Pπ]n (2π)n R Let  (f ∗g)(x) = f(y)g(x−y) (3.6) y∈V X denote convolution, and let f∗i denote the convolution of i factors of f. Recall from [2] that τˆ (k) ≥ 0 for all k. For i,j non-negative integers, let p T(i,j) = |Dˆ(k)|iτˆ (k)j, (3.7) p p Z T = sup(pΩ)(D ∗τ∗3)(x). (3.8) p p x We will use the following lemma, which provides minor extensions of results of [9, 15]. The lemma will also be useful in [18]. 5 Lemma 3.1. For G = Zn and G = Q , there are constants K and K such that for all p ≤ p¯ (G), n i,j c T(i,j) ≤ K Ω−i/2 (i,j ≥ 0), (3.9) p i,j T ≤ KΩ−1, (3.10) p KΩ−1 (i = 1) supτ(i)(x) ≤ (3.11) x p 2iKi,1Ω−i/2 (i ≥ 2). The above bounds are valid for n ≥ 1 for Q, and for n larger than an absolute constant for Zn, n except (3.9) also requires n ≥ 2j +1 for Zn. Proof. We prove the bounds (3.9)–(3.11) in sequence. Proof of (3.9). We first prove that for Zn and Q , and for positive integers i, there is a positive n a such that i a Dˆ(k)2i ≤ i. (3.12) Ωi Z The left side is equal to the probability that a random walk that starts at the origin returns to the origin after 2i steps, and therefore is equal to Ω−2i times the number of walks that make the transition from 0 to 0 in 2i steps. Each such walk must take an even number of steps in each coordinate direction, so it must lie within a subspace of dimension ℓ ≤ min{i,n}. If we fix the subspace, then each step in the subspace can be chosen from at most 2ℓ different directions (for Q , from ℓ directions). Thus, there are at most (2ℓ)2i walks in the subspace. Since the number of n subspaces of fixed dimension ℓ is given by n ≤ nℓ/ℓ!, we obtain the bound ℓ (cid:16) (cid:17) i 1 i 1 nℓ(2ℓ)2i ≤ nii2i 22i (3.13) ℓ! ℓ! ℓ=1 ℓ=1 X X for the number of walks that make the transition from 0 to 0 in 2i steps. Multiplying by Ω−2i to convert the number of walks into a probability leads to (3.12). This proves (3.9) for j = 0, so we take j ≥ 1. Fix an even integer s = s(j) such that t = s/(s−1) obeys jt < j+ 1. By H¨older’s inequality, 2 1/s 1/t T(i,j) ≤ Dˆ(k)is τˆ (k)jt . (3.14) p p (cid:18)Z (cid:19) (cid:18)Z (cid:19) By (3.12), it suffices to show that τˆ (k)jt is bounded by a constant depending on j. We give p separate arguments for this, for Zn and Q . R n For Zn, the infrared bound [15, (4.7)] implies that τˆ (k) ≤ 2[1−Dˆ(k)]−1 for sufficiently large p n, uniformly in p ≤ p (Zn). Thus, c 1 τˆ (k)jt ≤ 2jt . (3.15) p [1−Dˆ(k)]jt Z Z For A > 0 and m > 0, 1 1 ∞ = um−1e−uAdu, (3.16) Am Γ(m) Z0 6 so that 1 1 ∞ π dθ n = duujt−1 e−un−1(1−cosθ) . (3.17) Z [1−Dˆ(k)]jt Γ(jt) Z0 (cid:18)Z−π 2π(cid:19) The right side is non-increasing in n, since kfk ≤ kfk for 0 < p ≤ q ≤ ∞ on a probability space. p q Since n 2 |k|2 1−Dˆ(k) = (1−cosk ) ≥ , (3.18) j π2 n j=1 X and since 2jt < 2j +1, the integral on the left hand side of (3.17) is finite when n = 2j +1. This completes the proof for Zn. For Q , we use the fact that τˆ (0) = χ(p) to see that n p τˆ (k)jt = 2−nχ(p)jt +2−n τˆ (k)jt. (3.19) p p Z k∈{0,Xπ}n:k6=0 The first term on the right hand side is at most 2−nχ(p¯ (Q ))jt = 2−nfjt, which is exponentially c n n small. For the second term, we recall from [9, Theorem 6.1] that τˆ (k) ≤ [1+O(n−1)][1−Dˆ(k)]−1, p so it suffices to prove that 1 2−n (3.20) [1−Dˆ(k)]jt k∈{0,Xπ}n:k6=0 is bounded uniformly in n ≥ 1. For this, we let m(k) denote the number of nonzero components of k. We fix an ε > 0 and divide the sum according to whether m(k) ≤ εn or m(k) > εn. An elementary computation (see [9, Section 2.2.1]) gives 1 − Dˆ(k) = 2m(k)/n. Therefore, the contribution to (3.20) due to m(k) > εn is bounded by a constant depending only on ε and j. On the other hand, for k 6= 0, we use 1−Dˆ(k) = 2m(k)/n ≥ 2/n to see that 1 2−n ≤ 2−jtnjt2−n 1 [1−Dˆ(k)]jt k∈{0,π}nX:0<m(k)≤εn k∈{0,π}nX:0<m(k)≤εn εn n = 2−jtnjt2−n m! m=1 X ≤ 2−jtnjtP(X ≤ εn), (3.21) where X is a binomial random variable with parameters (n,1/2). Since E[X] = n/2, the right side of (3.21) is exponentially small in n as n → ∞ if we choose ε < 1, by standard large deviation 2 bounds for the binomial distribution (see, e.g., [5, Theorem A.1.1]). This completes the proof for Q . n Proof of (3.10). We repeat the argument of [9, Lemma 5.5] for Q , which applies verbatim for Zn. n It follows from the BK inequality that if x 6= 0 then τ (x) ≤ pΩ(D ∗τ )(x). (3.22) p p Using this, we conclude that pΩ(D ∗τ∗3)(x) ≤ pΩD(x)+3(pΩ)2(D∗2 ∗τ∗3)(x), (3.23) p p 7 where the first term is the contribution where each of the three two-point functions τ (u) in τ∗3 p p is evaluated at u = 0, and the second term takes into account the case where at least one of the three displacements is nonzero. Since p ≤ p¯ = Ω−1 +O(Ω−2) ≤ 2Ω−1 for large Ω, this gives c T ≤ 2Ω−1 +12T(2,3) ≤ (2+12K )Ω−1 = KΩ−1, (3.24) p p 2,3 where in the first inequality we used (3.4) to rewrite the second term of (3.23). Proof of (3.11). For i ≥ 1, the BK inequality can be applied as in the proof of (3.22) to obtain τ(i)(x) ≤ (pΩ)i(D∗i ∗τ )(x). (3.25) p p It follows from (3.4) and (3.25) that supτ(i)(x) ≤ sup(pΩ)i Dˆ(k)iτˆ (k)e−ik·x ≤ (pΩ)iT(i,1) ≤ 2iK Ω−i/2, (3.26) p p p i,1 x x Z where we have used the fact that pΩ ≤ 2 for Ω sufficiently large. For i = 1, this can be improved by observing that, for Ω sufficiently large, τ(1)(x) ≤ pΩD(x)+τ(2)(x) ≤ 2Ω−1 +2K Ω−1. (3.27) p p 2,1 4 Proof of Proposition 2.1 We now complete the proof of Proposition 2.1, by proving (2.8), (2.9), (2.10) in Sections 4.1, 4.2, 4.3, respectively. Throughout this section we fix p ≤ p¯ (G). c 4.1 Expansion for Πˆ(0) p Given a configuration, we say that x is doubly connected to y, and we write x ⇔ y, if x = y or if there are at least two bond-disjoint paths from x to y consisting of occupied bonds. For ℓ ≥ 4, an ℓ-cycle is a set of bonds that can be written as {{v ,v }} with v = v and otherwise v 6= v i−1 i 1≤i≤ℓ ℓ 0 i j for i 6= j, and a cycle is an ℓ-cycle for some ℓ ≥ 4. By definition, Πˆ(0) = P (0 ⇔ x) = P (∃ occupied cycle containing 0,x). (4.1) p p p x6=0 x6=0 X X We decompose the summand into (a) the probability that there exists an occupied 4-cycle contain- ing 0,x, plus (b) the probability that there exists an occupied cycle of length at least 6 containing 0,x and no occupied 4-cycle containing 0,x. The contribution to Πˆ(0) due to (a) is bounded above by summing p4 over x 6= 0 and over p 4-cycles containing 0,x. The number of 4-cycles containing 0 is 1ΩΩ′, and each such cycle has 2 three possibilities for x. Therefore 3 contribution due to (a) ≤ ΩΩ′p4. (4.2) 2 8 For a lower bound, we apply inclusion-exclusion and subtract from this upper bound the sum of p7 over x 6= 0 and over pairs of 4-cycles, each containing 0,x. In this case, x must be a neighbour of 0, and p7 is the probability of simultaneous occupation of the two 4-cycles. There are order Ω3 such pairs of 4-cycles. Since we already know that p¯ (G) ≤ O(Ω−1), this gives c 3 3 contribution due to (a) = ΩΩ′p4 +O(Ω3p7) = ΩΩ′p4 +O(Ω−4). (4.3) 2 2 For the contribution due to (b), we use Lemma 4.1 below. Given increasing events E,F, we use the standard notation E◦F to denote the event that E and F occur disjointly. Roughly speaking, E ◦F is the set of bond configurations for which there exist two disjoint sets of occupied bonds such that the first set guarantees the occurrence of E and the second guarantees the occurrence of F. The BK inequality asserts that P(E ◦F) ≤ P(E)P(F), for increasing events E and F. (See [14, Section 2.3] for a proof, and for a precise definition of E ◦F.) Lemma 4.1. Let p ≤ p¯ (G). Let Π(0,ℓ)(x) denote the probability that there is an occupied cycle c p containing 0,x, of length ℓ or longer. Then for ℓ ≥ 4 and for Ω sufficiently large (not depending on ℓ), Π(0,ℓ)(x) ≤ (ℓ−1)2ℓK Ω−ℓ/2. (4.4) p ℓ,2 x6=0 X Proof. Let ℓ ≥ 4, and suppose there exists an occupied cycle containing 0,x, of length ℓ or longer. Then there is a j ∈ {1,...,ℓ−1} such that {0 ←−→ x}◦{0 ←−−→ x} occurs. Therefore, by the BK j ℓ−j inequality, ℓ−1 Π(0,ℓ)(x) ≤ τ(j)(x)τ(ℓ−j)(x). (4.5) p p p j=1 X By (3.25), by the fact that pΩ ≤ 2 for Ω sufficiently large, and by (3.9), it follows that Π(0,ℓ)(x) ≤ (ℓ−1)2ℓ(D∗ℓ ∗τ∗2)(0) ≤ (ℓ−1)2ℓT(ℓ,2) ≤ (ℓ−1)2ℓK Ω−ℓ/2, (4.6) p p p ℓ,2 x6=0 X as required. The contribution due to case (b) is therefore at most Π(0,6)(x) ≤ O(Ω−3), and hence x6=0 p P 3 Πˆ(0) = ΩΩ′p4 +O(Ω−3), (4.7) p 2 which proves (2.8). 4.2 Expansion for Πˆ(1) p To define Πˆ(1), we need the following definitions. p Definition 4.2. (i) Given a bond configuration, vertices x,y, and a set A of vertices of G, we say A x and y are connected through A, and write x ↔ y, if every occupied path connecting x to y has at least one bond with an endpoint in A. (ii) Given a bond configuration, and a bond b, we define C˜b(x) to be the set of vertices connected to x in the new configuration obtained by setting b to be vacant. 9 (iii) Given a bond configuration and vertices x,y, we say that the directed bond (u,v) is pivotal for x ↔ y if (a) x ↔ y occurs when the bond {u,v} is set occupied, and (b) when {u,v} is set vacant x ↔ y does not occur, but x ↔ u and v ↔ y do occur. (Note that there is a distinction between the events {(u,v) is pivotal for x ↔ y} and {(v,u) is pivotal for x ↔ y} = {(u,v) is pivotal for y ↔ x}.) Let E′(v,x;A) = {v ↔A x}∩{6 ∃ pivotal (u′,v′) for v ↔ x s.t. v ↔A u′}. (4.8) We will refer to the “no pivotal” condition of the second event on the right hand side of (4.8) as the “NP” condition. By definition, Πˆ(1) = p E I[0 ⇔ u]P (E′(v,x;C˜(u,v)(0))) , (4.9) p 0 1 0 Xx (Xu,v) h i where thesumover (u,v)isa sumover directed bonds. Ontherighthandside, thecluster C˜(u,v)(0) 0 is random with respect to the expectation E , so that C˜(u,v)(0) should be regarded as a fixed set 0 0 inside the probability P . The latter introduces a second percolation model which depends on the 1 original percolation model via the set C˜(u,v)(0). We use subscripts for C˜ and the expectations, 0 to indicate to which expectation C˜ belongs, and refer to the bond configuration corresponding to expectation j as the “level-j” configuration. We also write F to indicate an event F at level-j. j Then (4.9) can be written as Πˆ(1) = p P(1) {0 ⇔ u} ∩E′(v,x;C˜(u,v)(0)) , (4.10) p 0 0 1 Xx (Xu,v) h i where P(1) represents the joint expectation of the percolation models at levels-0 and 1. We begin with a minor extension of a standard estimate for Πˆ(1) (see [9, Section 4] for related p ˜ ˜(u,v) discussion with our present notation). Making the abbreviation C = C (0), we may insert 0 0 within the square brackets on the right hand side of (4.10) the disjoint union • • {u = 0}∩{x ∈ C˜ } {u = 0}∩{x6∈C˜ } {u 6= 0}. (4.11) 0 0 (cid:16) (cid:17) [ (cid:16) (cid:17) [ The first term is the leading term and the other two produce error terms. We first show that the term {u 6= 0} produces an error term. We define the events F (0,u,w,z) = {0 ↔ u}◦{0 ↔ w}◦{w ↔ u}◦{w ↔ z}, (4.12) 0 F (v,t,z,x) = {v ↔ t}◦{t ↔ z}◦{t ↔ x}◦{z ↔ x}. (4.13) 1 Note that F (v,t,z,x) = F (x,z,t,v). Recalling the definition of {x ←−→ y} from (3.1), we also 1 0 j define F(j)(0,u,w,z) = {0 ←−→ u}◦{0 ←−→ w}◦{w ←−→ u}◦{w ↔ z}, (4.14) 0 j1+j[2+j3=j j1 j2 j3 F(j)(v,t,z,x) = {v ↔ t}◦{t ←−→ z}◦{t ←−→ x}◦{z ←−→ x}. (4.15) 1 j1+j[2+j3=j j1 j2 j3 10