Expansion in n 1 for percolation critical values on the n-cube and Z n : the first three terms

Transcription

1 Eansion in n for ercolation critical values on the n-cube and Z n : the first three terms Remco van der Hofstad Gordon Slade December 22, 2003 Abstract Let c (Q n ) and c (Z n ) denote the critical values for nearest-neighbour bond ercolation on the n-cube Q n = {0, } n and on Z n, resectively. Let Ω = n for G = Q n and Ω = 2n for G = Z n denote the degree of G. We use the lace eansion to rove that for both G = Q n and G = Z n, c (G) = Ω + Ω Ω 3 + O(Ω 4 ). This etends by two terms the result c (Q n ) = Ω + O(Ω 2 ) of Borgs, Chayes, van der Hofstad, Slade and Sencer, and rovides a simlified roof of a revious result of Hara and Slade for Z n. Main result We consider bond ercolation on Z n with edge set consisting of airs {, y} of vertices in Z n with y =, where w = n j= w j for w Z n. Bonds (edges) are indeendently occuied with robability and vacant with robability. We also consider bond ercolation on the n-cube Q n, which has verte set {0, } n and edge set consisting of airs {, y} of vertices in {0, } n with y =, where we regard Q n as an additive grou with addition comonent-wise modulo 2. Again bonds are indeendently occuied with robability and vacant with robability. We write G in lace of Q n and Z n when we wish to refer to both models simultaneously. We write Ω for the degree of G, so that Ω = 2n for Z n and Ω = n for Q n. For the case of Z n, the critical value is defined by c (Z n ) = inf{ : an infinite connected cluster of occuied bonds a.s.}. (.) Given a verte of G, let C() denote the connected cluster of, i.e., the set of vertices y such that y is connected to by a ath consisting of occuied bonds. Let C() denote the cardinality Deartment of Mathematics and Comuter Science, Eindhoven University of Technology, P.O. Bo 53, 5600 MB Eindhoven, The Netherlands. rhofstad@win.tue.nl Deartment of Mathematics, University of British Columbia, Vancouver, BC V6T Z2, Canada. slade@math.ubc.ca

2 of C(), and let χ() = E C(0) denote the eected cluster size of the origin. Results of [, 20] imly that c (Z n ) = su{ : χ() < }. (.2) is an equivalent definition of the critical value. For ercolation on a finite grah G, such as Q n, the above characterizations of c (G) are inalicable. In [8, 9, 0] (in articular, see [0]), it was shown that there is a small ositive constant λ 0 such that the critical value c (Q n ) = c (Q n ; λ 0 ) for the n-cube is defined imlicitly by χ( c (Q n )) = λ 0 2 n/3. (.3) Given λ 0, (.3) uniquely secifies c (Q n ), since χ() is a olynomial in that increases from χ(0) = to χ() = 2 n. Our main result is the following theorem. Theorem.. (i) For G = Z n, c (Z n ) = 2n + (2n) (2n) + ) 3 O( (2n) 4 as n. (.4) (ii) For Q n, fi constants c, c indeendent of n, and choose such that χ() [cn 3, c n 6 2 n ] (e.g., = c (Q n ; λ 0 )). Then = n + n n + ) 3 O( n 4 as n. (.5) The constant in the error term deends on c, c, but does not deend otherwise on. By Theorem., the eansions of c (G) in owers of Ω are the same for Q n and Z n, u to and including order Ω 3. Higher order coefficients could be comuted using our methods, but the labour cost increases sharly with each subsequent term. Although we sto short of comuting the coefficient of Ω 4, we eect that the coefficients for Q n and Z n will differ at this order. In [8], for both Q n and Z n, we rove the eistence of asymtotic eansions for c (G) to all orders in Ω, without comuting the numerical values of the coefficients. For Q n, it was shown by Ajtai, Komlós and Szemerédi [3] that c (Q n ) > n (+ɛ) for every fied ɛ > 0 (although the above definition of c (Q n ) did not aear until [8]). Bollobás, Kohayakawa and Luczak [7] imroved this to c (Q n ) [ e o(n), (log n) ]. Theorem. etends the very n n n 2 recent result c (Q n ) = n + O(n 2 ) of [8, 9] by two terms. Bollobás, Kohayakawa and Luczak [7] raised the question of whether the critical value might be equal to, but we see from (.5) that n c (Q n ) = + 5 n 2 n 3 + O(n 4 ). For Z n, Theorem. is identical to a result of Hara and Slade [6, 7]. Earlier, Bollobás and Kohayakawa [6], Gordon [3], Kesten [9] and Hara and Slade [5] obtained the first term in (.4) for Z n with error terms O((log n) 2 n 2 ), O(n 65/64 ), O((log log n) 2 (n log n) ) and O(n 2 ), resectively. Recently, Alon, Benjamini and Stacey [4] gave an alternate roof that c (Z n ) is asymtotic to (2n) as n. The eansion c (Z n ) = 2n + (2n) (2n) (2n) (2n) 5 + (.6) 2

3 was reorted in [2], but with no rigorous bound on the remainder. We remark that for oriented ercolation on Z n, defined in such a way that the forward degree is n, it was roved in [] that the critical value obeys the bounds n + ( ) 2 n + o 3 n 3 c (oriented Z n ) n + ( ) n + O. (.7) 3 n 4 Our method is based on the lace eansion and alies the general aroach of [6, 7] that was used to rove Theorem.(i) for Z n, but our method here is simler and alies to Z n and Q n simultaneously. Remark. For Q n, it is a direct consequence of [8, Proosition.2] that if there is some sequence (deending on n) with χ() [cn 3, c n 6 2 n ] such that = n + n n 3 + O(n 4 ), then the same asymtotic formula holds for all such. Thus it suffices to rove (.5) for a single such sequence. We fi some sequence f n such that lim n f n n M = for every ositive integer M and such that lim n f n e αn = 0 for every α > 0. We define by χ( ) = f n, and observe that eventually χ( ) [cn 3, c n 6 2 n ]. For G = Q n, it therefore suffices to rove that has the eansion (.5). We will use the notation (G = Q n ), c = c (G) = c (Z n ) (G = Z n ). 2 Alication of the lace eansion For Q n or Z n with n large, the lace eansion [5] gives rise to an identity (.8) χ() = + ˆΠ Ω[ + ˆΠ ], (2.) where ˆΠ is a function that is finite for c (G). Although we do not dislay the deendence elicitly in the notation, ˆΠ does deend on the grah Q n or Z n. The identity (2.) is valid for c (G). For a derivation of the lace eansion, see, e.g., [9, Section 3]. It follows from (2.) that Ω = + ˆΠ χ(). (2.2) The function ˆΠ has the form with (recall (.8)) ˆΠ = ( ) N ˆΠ(N), (2.3) N=0 ( ) C N ˆΠ (N) uniformly in c. (2.4) Ω For Q n, the formula (2.) and the bounds (2.4) are given in [9, (6.)] and [9, Lemma 5.4], resectively (with our ˆΠ written as ˆΠ (N) (0)). In more detail, [9, Lemma 5.4] states that ˆΠ [const(λ 3 β)] N, where λ = χ()2 n/3 f n 2 n/3 for c (Q n ). By definition, f n 2 n/3 is 3

4 eonentially small in n. In addition, it is shown in [9, Proosition 2.] that β can be chosen roortional to n. It follows from (2.2) that n c (Q n ) = + ˆΠ c(q n) + O(f n ). (2.5) The second term on the right hand side of (2.5) can be neglected in the roof of Theorem.. Equations (2.3) (2.5) give c (Q n ) = n + O(n 2 ). For Z n, (2.) and (2.4) follow from results in [5, Section 4.3.2]. (Note the notational difference (N) that in [5] what we are calling here ˆΠ is called ĝ N (0) and that Since χ( c (Z n )) =, it follows from (2.2) that 2n c (Z n ) = ˆΠ (N) in [5] is something different.) + ˆΠ. (2.6) c (Z n ) With (2.3) (2.4), this imlies that c (Z n ) = (2n) + O(n 2 ). The identities (2.5) and (2.6) give recursive equations for c. To rove Theorem. using this recursion, we will aly the following roosition. In its statement, we write Ω n for Q n = (2.7) 2n 2 for Z n. Proosition 2.. For G = Z n and G = Q n, uniformly in c (G), N=3 ˆΠ (0) = 3 2 ΩΩ 4 + O(Ω 3 ), (2.8) ˆΠ () = Ω 2 + 4ΩΩ 4 + O(Ω 3 ), (2.9) ˆΠ (2) = Ω 3 + Ω(Ω ) 4 + O(Ω 3 ), (2.0) ˆΠ (N) = O(Ω 3 ). (2.) We show now that Proosition 2. imlies Theorem.. It follows from Ω c (G) = + O(Ω ) (as noted below (2.5) and (2.6)), (2.3), and Proosition 2. that ˆΠ c (G) = Ω + O(Ω 2 ). (2.2) With (2.5) (2.6), this imlies that Ω c (G) = + Ω + O(Ω 2 ). (2.3) Using this in the bounds of Proosition 2., along with (2.3), gives ˆΠ c(g) = 3 2Ω 2 Ω( Ω + Ω 2 )2 4 Ω 2 + Ω 2 + Ω 2 + O(Ω 3 ) = Ω 5 2Ω 2 + O(Ω 3 ). (2.4) 4

5 Substitution of this imrovement of (2.2) into (2.5) (2.6) then gives Ω c (G) = + Ω + 7 2Ω 2 + O(Ω 3 ). (2.5) Thus, to rove Theorem., it suffices to rove Proosition 2.. Since (2.) is a consequence (N) of (2.4), we must rove (2.8) (2.0). Precise definitions of ˆΠ, for N = 0,, 2, will be given in Section 4. 3 Preliminaries Before roving Proosition 2., we recall and etend some estimates from [9, 5]. Let D() = Ω if is adjacent to 0, and D() = 0 otherwise. Thus D(y ) is the transition robability for simle random walk on G to make a ste from to y. Let τ (y ) = P ( y) denote the two-oint function. For i 0, we denote by { i y} (3.) the event that is connected to y by an occuied (self-avoiding) ath of length at least i, and define τ (i) (, y) = P( y). (3.2) i We define the Fourier transform of an absolutely summable function f on the verte set V of G by ˆf(k) = V f()e ik (k V ), (3.3) where V = {0, π} n for Q n and V = [ π, π] n for Z n. We write the inverse Fourier transform as f() = ˆf(k)e ik, (3.4) where we use the convenient notation 2 n k {0,π} ĝ(k) = ĝ(k) (G = Q n) n [ π,π] ĝ(k) n dn k (G = Z n ). (2π) n Let (3.5) (f g)() = y V f(y)g( y) (3.6) denote convolution, and let f i denote the convolution of i factors of f. Recall from [2] that ˆτ (k) 0 for all k. For i, j non-negative integers, let T (i,j) = ˆD(k) iˆτ (k) j, (3.7) T = su(ω)(d τ 3 )(). (3.8) We will use the following lemma, which rovides minor etensions of results of [9, 5]. The lemma will also be useful in [8]. 5

6 Lemma 3.. For G = Z n and G = Q n, there are constants K i,j and K such that for all c (G), su T (i,j) K i,j Ω i/2 (i, j 0), (3.9) T KΩ, (3.0) τ (i) () KΩ (i = ) 2 K i, Ω i/2 (i 2). (3.) The above bounds are valid for n for Q n, and for n larger than an absolute constant for Z n, ecet (3.9) also requires n 2j + for Z n. Proof. We rove the bounds (3.9) (3.) in sequence. Proof of (3.9). We first rove that for Z n and Q n, and for ositive integers i, there is a ositive a i such that ˆD(k) 2i a i Ω. (3.2) i The left side is equal to the robability that a random walk that starts at the origin returns to the origin after 2i stes, and therefore is equal to Ω 2i times the number of walks that make the transition from 0 to 0 in 2i stes. Each such walk must take an even number of stes in each coordinate direction, so it must lie within a subsace of dimension l min{i, n}. If we fi the subsace, then each ste in the subsace can be chosen from at most 2l different directions (for Q n, from l directions). Thus, there are at most (2l) 2i walks in the subsace. Since the number of subsaces of fied dimension l is given by ( ) n l n l /l!, we obtain the bound i l= l! nl (2l) 2i n i i 2i i l= l! 22i (3.3) for the number of walks that make the transition from 0 to 0 in 2i stes. Multilying by Ω 2i to convert the number of walks into a robability leads to (3.2). This roves (3.9) for j = 0, so we take j. Fi an even integer s = s(j) such that t = s/(s ) obeys jt < j +. By Hölder s inequality, 2 T (i,j) ( /s ( ˆD(k) is) ˆτ (k) jt ) /t. (3.4) By (3.2), it suffices to show that ˆτ (k) jt is bounded by a constant deending on j. We give searate arguments for this, for Z n and Q n. For Z n, the infrared bound [5, (4.7)] imlies that ˆτ (k) 2[ ˆD(k)] for sufficiently large n, uniformly in c (Z n ). Thus, ˆτ (k) jt 2 jt. (3.5) [ ˆD(k)] jt For A > 0 and m > 0, A = u m e ua du, (3.6) m Γ(m) 0 6

7 so that = [ ˆD(k)] jt Γ(jt) 0 du u jt ( π π e un ( cos θ) dθ ) n. (3.7) 2π The right side is non-increasing in n, since f f q for 0 < q on a robability sace. Since ˆD(k) n = ( cos k j ) 2 k 2 π 2 n, (3.8) j= and since 2jt < 2j +, the integral on the left hand side of (3.7) is finite when n = 2j +. This comletes the roof for Z n. For Q n, we use the fact that ˆτ (0) = χ() to see that ˆτ (k) jt = 2 n χ() jt + 2 n k {0,π} n :k 0 ˆτ (k) jt. (3.9) The first term on the right hand side is at most 2 n χ( c (Q n )) jt = 2 n fn jt, which is eonentially small. For the second term, we recall from [9, Theorem 6.] that ˆτ (k) [ + O(n )][ ˆD(k)], so it suffices to rove that 2 n [ ˆD(k)] (3.20) jt k {0,π} n :k 0 is bounded uniformly in n. For this, we let m(k) denote the number of nonzero comonents of k. We fi an ε > 0 and divide the sum according to whether m(k) εn or m(k) > εn. An elementary comutation (see [9, Section 2.2.]) gives ˆD(k) = 2m(k)/n. Therefore, the contribution to (3.20) due to m(k) > εn is bounded by a constant deending only on ε and j. On the other hand, for k 0, we use ˆD(k) = 2m(k)/n 2/n to see that 2 n k {0,π} n :0<m(k) εn [ ˆD(k)] jt 2 jt n jt 2 n = 2 jt n jt 2 n εn k {0,π} n :0<m(k) εn ) m= ( n m 2 jt n jt P(X εn), (3.2) where X is a binomial random variable with arameters (n, /2). Since E[X] = n/2, the right side of (3.2) is eonentially small in n as n if we choose ε <, by standard large deviation 2 bounds for the binomial distribution (see, e.g., [5, Theorem A..]). This comletes the roof for Q n. Proof of (3.0). We reeat the argument of [9, Lemma 5.5] for Q n, which alies verbatim for Z n. It follows from the BK inequality that if 0 then Using this, we conclude that τ () Ω(D τ )(). (3.22) Ω(D τ 3 )() ΩD() + 3(Ω) 2 (D 2 τ 3 )(), (3.23) 7

8 where the first term is the contribution where each of the three two-oint functions τ (u) in τ 3 is evaluated at u = 0, and the second term takes into account the case where at least one of the three dislacements is nonzero. Since c = Ω + O(Ω 2 ) 2Ω for large Ω, this gives T 2Ω + 2T (2,3) (2 + 2K 2,3 )Ω = KΩ, (3.24) where in the first inequality we used (3.4) to rewrite the second term of (3.23). Proof of (3.). For i, the BK inequality can be alied as in the roof of (3.22) to obtain τ (i) () (Ω) i (D i τ )(). (3.25) It follows from (3.4) and (3.25) that su τ (i) () su(ω) i ˆD(k)iˆτ (k)e ik (Ω) i T (i,) 2 i K i, Ω i/2, (3.26) where we have used the fact that Ω 2 for Ω sufficiently large. For i =, this can be imroved by observing that, for Ω sufficiently large, τ () () ΩD() + τ (2) () 2Ω + 2K 2, Ω. (3.27) 4 Proof of Proosition 2. We now comlete the roof of Proosition 2., by roving (2.8), (2.9), (2.0) in Sections 4., 4.2, 4.3, resectively. Throughout this section we fi c (G). 4. Eansion for ˆΠ (0) Given a configuration, we say that is doubly connected to y, and we write y, if = y or if there are at least two bond-disjoint aths from to y consisting of occuied bonds. For l 4, an l-cycle is a set of bonds that can be written as {{v i, v i }} i l with v l = v 0 and otherwise v i v j for i j, and a cycle is an l-cycle for some l 4. By definition, ˆΠ (0) = P (0 ) = P ( occuied cycle containing 0, ). (4.) 0 0 We decomose the summand into (a) the robability that there eists an occuied 4-cycle containing 0,, lus (b) the robability that there eists an occuied cycle of length at least 6 containing 0, and no occuied 4-cycle containing 0,. ˆΠ (0) The contribution to due to (a) is bounded above by summing 4 over 0 and over 4-cycles containing 0,. The number of 4-cycles containing 0 is 2 ΩΩ, and each such cycle has three ossibilities for. Therefore contribution due to (a) 3 2 ΩΩ 4. (4.2) 8

9 For a lower bound, we aly inclusion-eclusion and subtract from this uer bound the sum of 7 over 0 and over airs of 4-cycles, each containing 0,. In this case, must be a neighbour of 0, and 7 is the robability of simultaneous occuation of the two 4-cycles. There are order Ω 3 such airs of 4-cycles. Since we already know that c (G) O(Ω ), this gives contribution due to (a) = 3 2 ΩΩ 4 + O(Ω 3 7 ) = 3 2 ΩΩ 4 + O(Ω 4 ). (4.3) For the contribution due to (b), we use Lemma 4. below. Given increasing events E, F, we use the standard notation E F to denote the event that E and F occur disjointly. Roughly seaking, E F is the set of bond configurations for which there eist two disjoint sets of occuied 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 [4, Section 2.3] for a roof, and for a recise definition of E F.) Lemma 4.. Let c (G). Let Π (0,l) () denote the robability that there is an occuied cycle containing 0,, of length l or longer. Then for l 4 and for Ω sufficiently large (not deending on l), Π (0,l) () (l )2 l K l,2 Ω l/2. (4.4) 0 Proof. Let l 4, and suose there eists an occuied cycle containing 0,, of length l or longer. Then there is a j {,..., l } such that {0 } {0 } occurs. Therefore, by the BK j l j inequality, l Π (0,l) () τ (j) ()τ (l j) (). (4.5) By (3.25), by the fact that Ω 2 for Ω sufficiently large, and by (3.9), it follows that 0 Π (0,l) j= () (l )2 l (D l τ 2 )(0) (l )2 l T (l,2) (l )2 l K l,2 Ω l/2, (4.6) as required. The contribution due to case (b) is therefore at most 0 Π (0,6) () O(Ω 3 ), and hence which roves (2.8). 4.2 Eansion for To define ˆΠ () ˆΠ (), we need the following definitions. ˆΠ (0) = 3 2 ΩΩ 4 + O(Ω 3 ), (4.7) Definition 4.2. (i) Given a bond configuration, vertices, y, and a set A of vertices of G, we say and y are connected through A, and write A y, if every occuied ath connecting to y has at least one bond with an endoint in A. (ii) Given a bond configuration, and a bond b, we define C b () to be the set of vertices connected to in the new configuration obtained by setting b to be vacant. 9

10 (iii) Given a bond configuration and vertices, y, we say that the directed bond (u, v) is ivotal for y if (a) y occurs when the bond {u, v} is set occuied, and (b) when {u, v} is set vacant y does not occur, but u and v y do occur. (Note that there is a distinction between the events {(u, v) is ivotal for y} and {(v, u) is ivotal for y} = {(u, v) is ivotal for y }.) Let E (v, ; A) = {v A } { ivotal (u, v ) for v s.t. v A u }. (4.8) We will refer to the no ivotal condition of the second event on the right hand side of (4.8) as the NP condition. By definition, ˆΠ () = [ E 0 I[0 u]p (E (u,v) (v, ; C 0 (0))) ], (4.9) (u,v) where the sum over (u, v) is a sum over directed bonds. On the right hand side, the cluster 0 (0) (u,v) is random with resect to the eectation E 0, so that C 0 (0) should be regarded as a fied set inside the robability P. The latter introduces a second ercolation model which deends on the (u,v) original ercolation model via the set C 0 (0). We use subscrits for C and the eectations, to indicate to which eectation C belongs, and refer to the bond configuration corresonding to eectation j as the level-j configuration. We also write F j to indicate an event F at level-j. Then (4.9) can be written as ˆΠ () = [ P () {0 u}0 E (v, ; (u,v) C (u,v) C (u,v) 0 (0)) ], (4.0) where P () reresents the joint eectation of the ercolation models at levels-0 and. () We begin with a minor etension of a standard estimate for ˆΠ (see [9, Section 4] for related discussion with our resent notation). Making the abbreviation C (u,v) 0 = C 0 (0), we may insert within the square brackets on the right hand side of (4.0) the disjoint union ( {u = 0} { C0 } ) ( {u = 0} { C0 } ) {u 0}. (4.) The first term is the leading term and the other two roduce error terms. We first show that the term {u 0} roduces an error term. We define the events F 0 (0, u, w, z) = {0 u} {0 w} {w u} {w z}, (4.2) F (v, t, z, ) = {v t} {t z} {t } {z }. (4.3) Note that F (v, t, z, ) = F 0 (, z, t, v). Recalling the definition of { j define F (j) 0 (0, u, w, z) = F (j) (v, t, z, ) = j +j 2 +j 3 =j j +j 2 +j 3 =j y} from (3.), we also {0 j u} {0 j2 w} {w j3 u} {w z}, (4.4) {v t} {t j z} {t j2 } {z j3 }. (4.5) 0

11 For u 0, it can be seen from the fact that u and 0 are in a level-0 cycle of length at least 4 that {0 u 0} 0 E (v, ; C 0 ) ( F (4) 0 (0, u, w, z) 0 F (v, t, z, ) ), (4.6) and hence this contribution to Let ˆΠ (),(u,v),t,w,z is at most t,w,z P (F (4) 0 (0, u, w, z))p (F (v, t, z, )). (4.7) A 3 (t, z, ) = τ ( t)τ (z t)τ (z ), (4.8) A (j) 3 (t, z, ) = τ (j ) ( t)τ (j 2 ) (z t)τ (j 3 ) (z ), (4.9) j +j 2 +j 3 =j B (w, u, z, t) = (ΩD τ )(t u)τ (z w). (4.20) By the BK inequality, (4.7) is at most u,w A (4) 3 (0, u, w) t,z B (w, u, z, t) A 3 (t, z, ). (4.2) Relacing w, z, t, by w = w + u, z = z + u, t = t + u, = + u, and using symmetry, this is equal to u,w A (4) 3 (0, u, w ) t,z B (w, 0, z, t ) A 3 (t, z, ). (4.22) We note that B (w, 0, z, t ) = B ( t, t, z w t, 0), and set z = z t, = t and then t = t to rewrite (4.22) as 3 (0, u, w ) B (t, t, z w, 0) A 3 (0, z, ) t,z ( ) A (4) 3 (0, u, w ) su B (t, t, a, 0) A 3 (0, z, ). (4.23) u,w a t z, u,w A (4) By (3.8) and the fact that τ (u) (τ τ )(u), Also, su a t and, using (3.25) and Ω 2, u,w A (4) B (t, t, a, 0) = su(ωd τ τ )(a) T. (4.24) a z, A 3 (0, z, ) = (τ τ τ )(0) = T (0,3), (4.25) 3 (0, u, w ) = (τ (j ) j +j 2 +j 3 =4 τ (j 2 ) τ (j 3 ) )(0) O(T (4,3) ). (4.26)

12 0 0 v Figure : Deiction of the event aearing in (4.28). Line 0 corresonds to a connection in level-0 and line to a connection in level-. Therefore, (4.23) is bounded above by O(T (4,3) T T (0,3) ), which is O(Ω 3 ) by (3.9) and (3.0). Similarly, an uer bound O(Ω 3 ) can be obtained for the contribution due to {u = 0} { C 0 }, starting from the observation that {u = 0} { C 0 } E (v, ; C 0 ) ( F 0 (0, 0, 0, z) 0 F (4) (v, t, z, ) ). (4.27) t,w,z:z The inclusion (4.27) follows from the fact that if C 0, then to obtain a non-zero contribution to P (E (v, ; C 0 )), must be in a level- occuied cycle of length at least 4 which contains a verte z C 0. We are left to consider the leading term ( P () { C(0,v) 0 (0)} E (v, ; (0,v) C (0,v) 0 (0)) ). (4.28) See Figure for a deiction of the event aearing in (4.28). The event in (4.28) is a subset of the event { C 0 } {v }. Thus, either there is a level-0 connection from 0 to (not using the bond {0, v} 0 ) of length l 0 and a level- connection from v to of length l, with l 0 + l 4, or {0 i0 } 0 {v i } occurs with i 0 + i = 5. This decomosition is not disjoint, as the latter ossibility does not imly that the former does not occur, but this is fine for an uer bound. By (3.25) and (3.9), the contribution due to the latter case is bounded above by 5 5 τ (i 0 ) ()τ (5 i 0 ) ( v) = (ΩD τ (i 0 ) τ (5 i 0 ) )(0) i 0 =0 (0,v) i 0 =0 6(Ω) 6 T (6,2) = O(Ω 3 ), (4.29) so this is an error term. Since v and 0 have oosite arity, if there is a level-0 connection from 0 to of length l 0 and a level- connection from v to of length l, then l 0 + l must be odd. Thus, we are left to deal with the cases l 0 + l = and l 0 + l = 3, and we consider these searately. (0,v) Case that l 0 + l =. If l = 0, then = v C 0 (0), which forces l 0 3. This is inconsistent with l 0 + l = and therefore need not be considered here. We may therefore assume that l 0 = 0 (0,v) and l =, so that = 0, {0, v} is occuied, and, to satisfy the NP condition of (4.8), v C 0 (0). We use inclusion-eclusion on the latter, writing I[v C (0,v) 0 (0)] = I[v 2 C (0,v) 0 (0)]. (4.30)

13 The first term contributes (0,v) = Ω 2. (4.3) The second term requires a level-0 connection from 0 to v of length 3 or more, which has robability (v), so that by (3.25) and (3.9), the second term contributes τ (3) (0,v) τ (3) (v) = (Ω)(D τ (3) )(0) (Ω) 4 T (4,) = O(Ω 3 ), (4.32) and hence is an error term. Thus, the case l 0 + l = contributes Ω 2 + O(Ω 3 ). (4.33) Case that l 0 + l = 3. There are four ossibilities: l = 0,, 2, 3. If l = 0 then = v, the NP condition is trivially satisfied, and there is an occuied level-0 ath from 0 to v of length 3. This contribution is ΩΩ 4 + O(Ω 3 7 ) = ΩΩ 4 + O(Ω 4 ), (4.34) where we have used inclusion-eclusion in a manner similar to that of the argument around (4.2) (4.3). In more detail, the first term in (4.34) accounts for the sum of the robability of an occuied level-0 ath of length 3 from 0 to v, while the second term accounts for overcounting due to simultaneous occuation of more than one such ath. For l =, 2, 3, we note that { C 0 } E (v, ; C 0 ) = { C 0 } {v } NP, (4.35) and use I[NP] = I[NP c ], to conclude that I[ C 0 ]I[E (v, ; C 0 ) ] = I[ C 0 ]I[{v } ] I[ C 0 ]I[{v } ]I[NP c ]. (4.36) We first consider the first term on the right hand side of (4.36). In the following, we write e to denote a neighbour of 0 that is not ±v, and which will ultimately be summed over. We again aly an inclusion-eclusion argument similar to that used for (4.34), but do not discuss its details. The case l = corresonds to l 0 = 2, so that = v + e, with the three bonds {0, e} 0, {e, } 0, {, v} each occuied. This contributes ΩΩ 4. Note that in the related configuration in which {0, v} 0, {v, } 0, {, v} are each occuied, the level-0 ath {0, v} 0, {v, } 0 from 0 to uses the bond {0, v} 0, and therefore need not be considered. For Z n, the configuration with = 2v and with {0, v} 0, {v, 2v} 0, {v, 2v} each occuied need not be considered for the same reason. (Also, it contributes O(Ω 4 ) = O(Ω 3 ) which is an error term.) The case l = 2 corresonds to l 0 =, so that = e, either with the three bonds {0, } 0, {, +v}, {+v, v} each occuied, or with the three bonds {0, } 0, {0, }, {0, v} each occuied. This contributes 2ΩΩ 4. For Z n, the configuration with = v and with {0, v} 0, {0, v}, {0, v} each occuied contributes O(Ω 4 ) = O(Ω 3 ) and thus is an error term. The case l = 3 corresonds to l 0 = 0, so that = 0, with the three bonds {0, e}, {e, e + v}, {e + v, v} each occuied. This contributes ΩΩ 4. In summary, the first term on the right hand side of (4.36), with l =, 2, 3, contributes 5ΩΩ 4 + O(Ω 3 ). (4.37) 3

14 Net, we consider the effect of the second term in (4.36), for l =, 2, 3. For l =, we have seen above that, to leading order, {0, e} 0, {e, } 0, {, v} are each occuied. The only ossible ivotal bond for the level- connection from v to is therefore (v, ), and thus the failure of NP requires v C 0. This requires a level-0 connection, disjoint from the bonds {0, e} 0 and {e, } 0, which joins either 0 to v, e to v, or to v. This adds an additional factor O(Ω ) and hence roduces an error term. For l = 2, we have seen above that there are two cases to consider. Suose first that {0, = e} 0, {, + v}, { + v, v} are each occuied. The only ossible ivotal bonds for the level- connection from v to are (v, + v) and ( + v, ). Violation of NP therefore requires either (v, + v) is ivotal and v C 0, or ( + v, v) is ivotal and + v C 0. In either of these cases, the condition that C 0 contain an additional verte is a higher order effect and leads to an error term O(Ω 3 ). The remaining case for l = 2 has {0, = e} 0, {0, }, {0, v} each occuied. The only ossible ivotal bonds for the level- connection from v to are (v, 0) and (0, ). Violation of NP therefore requires either (v, 0) is ivotal and v C 0, or (0, ) is ivotal and 0 C 0. The first of these cases leads to an error term as above. For the second case, 0 C 0 is automatic, and inclusioneclusion alied to the requirement that (0, v) is ivotal leads to a net contribution for l = 2 of ΩΩ 4 + O(Ω 3 ). Finally, we consider l = 3. In this case, = 0, and {0, e}, {e, e + v}, {e + v, v} are each occuied. The only ossible violations of NP are: (v, v + e) is ivotal for the connection from v to = 0 and v C 0, or (v + e, e) is ivotal and v + e C 0, or (e, 0) is ivotal and e C 0. In any of these three cases, the condition that C 0 must contain the additional verte requires etra connections that roduce an error term O(Ω 3 ) overall, using reasoning analogous to that emloyed above. We have thus shown that the case l 0 + l = 3 yields a net contribution 5ΩΩ 4 ΩΩ 4 + O(Ω 3 ) = 4ΩΩ 4 + O(Ω 3 ). (4.38) In summary, combining (4.33) and (4.38), we have roved (2.9), namely 4.3 Eansion for By definition, ˆΠ (2) = ˆΠ (2) (u 0,v 0 ) (u,v ) ˆΠ () = Ω 2 + 4ΩΩ 4 + O(Ω 3 ). (4.39) 2 E 0 [I[0 u 0 ]E [ I[E (v 0, u ; C 0 )]E 2 I[E (v, ; C )] ]], (4.40) where we have made the abbreviations C 0 = C (u 0,v 0 ) 0 (0) and C = C (u,v ) (v 0 ). A standard estimate (2) for ˆΠ is (2) 0 ˆΠ 2T (0,3) (T T (0,3) ) 2 (4.4) (see, e.g., [9, Section 4.2]; one factor 2 in [9, Proosition 4.] is easily droed for N = 2). This (2) estimate arises from the uer bound for ˆΠ deicted in Figure 2. The factor 2 is due to the fact that there are two terms in the uer bound. The two factors T in each term arise from the two 4

15 u 0 v 0 u 0 v u v u v Figure 2: The standard diagrams bounding Π (2). All vertices other than 0 are summed over the verte set V of G, lines reresent factors of τ, and vertical bars reresent factors ΩD z 0 v 0 u v 0 Figure 3: Diagrammatic reresentation of the event (4.43). Line 0 corresonds to a connection in level-0, lines, 2, 3 corresond to connections in level- and line 4 to a connection in level-2. diagram loos containing lines with vertical bars, and the three factors T (0,3) arise from the other three diagram loos. (2) We claim that contributions to ˆΠ in which u 0 0, or u C 0, or C roduce an error term of order O(Ω 4 ). This follows from routine estimates, along the lines of those used in Section 4.2 to conclude that we could assume there that u = 0 and C 0. These estimates, which we do not write down here in detail, show for eamle that if u 0 0, then the factor T (0,3), arising from the leftmost diagram loo can be relaced by a constant multile of T (4,3). By (3.9) and (3.0), this leads to a bound O(Ω 4 ), which is an error term. Similarly, if u C (0,v 0) 0 (0), then the event E (v 0, u ; C (0,v 0) 0 (0)) requires that u must be in an occuied level- cycle of length at least 4. In this case, we may again use standard estimates to relace a factor T (0,3) in (4.4), arising from the diagram loo in Figure 2 containing u, by a constant multile of T (4,3), and again this contribution is O(Ω 4 ). Finally, the same situation arises when C (u,v ) 0 (v 0 ), in which case we can relace the factor T (0,3) arising from the rightmost diagram loo by a constant multile of T (4,3), and again this contribution is O(Ω 4 ). Thus, we are now left to analyze ( 2 P (2) {u C 0 } { C } E (v 0, u ; C 0 ) E (v, ; C ) 2 ), (4.42) (0,v 0 ) (u,v ) where we write P (2) for the joint robability of levels 0, and 2. Let G denote the intersection of events on the right hand side of (4.42). The event G on the right hand side of (4.42) is contained in the event {0 u } 0 { {v0 z} {z v } {z } } {v } 2, (4.43) z G which is deicted in Figure 3. For any choice of j 0, j, j 2, j 3, j 4, a subset of (4.43) is the event {0 u } 0 { {v0 z} {z v } {z } } {v } 2. (4.44) j0 j j2 j3 j4 z G 5

16 Since v 0 has odd arity, and since u and v have oosite arity, we may assume that j 0 + j + j 2 and j 2 + j 3 + j 4 are both odd. If j 0 + j + j 2 3, then a standard diagrammatic estimate gives O(T (4,3) T ) = O(Ω 3 ) for the contribution of (4.44). Similarly, if j 2 + j 3 + j 4 3, then again a standard diagrammatic estimate gives an uer bound O(T T (4,3) ) = O(Ω 3 ). Note that if u 0, then we may assume that j 0 + j + j 2 3, which gives an error term. Thus, we may assume that G occurs, that u = 0, and that there is a z such that the connections of Figure 3 occur with lines of length l 0 = 0, l, l 2, l 3, l 4, where This gives three ossibilities for (l 0, l, l 2, l 3, l 4 ), namely l + l 2 =, (4.45) l 2 + l 3 + l 4 =. (4.46) (0, 0,, 0, 0), (0,, 0,, 0), (0,, 0, 0, ), (4.47) and it suffices to comute the contribution from each of these cases. Case of (0, 0,, 0, 0). In this case, u 0 = u = 0, z = = v = v 0, and the bond {v 0, u } is occuied. We eamine the constraints imosed by the event G of (4.42). The events {u C 0 } and { C } occur trivially. For the event E (v 0, u ; C C0 C0 0 ), we note that {v 0 u } = {v 0 0} occurs. Violation of the NP condition requires {0 v 0 } 0, and this contributes at most (0,v 3 0 ) 3 τ (3) (v 0 ) 2 (Ω) 4 T (4,) O(Ω 4 ). Thus, u to an error term, we may assume that E (v 0, u ; C 0 ) occurs. Finally, the event E (v, ; C ) 2 occurs trivially, since v = C. This case contributes Ω 3 + O(Ω 3 ). (4.48) Case of (0,, 0,, 0). In this case, u = 0, z = u, v =. Also, the fact that C imlies that there must be an occuied level- ath from v 0 to z = u to = v that does not use the bond (u, v ). This imlies that the event {u v } occurs, and hence this case contributes an error 3 term because it corresonds to (4.44) with j 3 = 3. Case of (0,, 0, 0, ). In this case, = z = u = u 0 = 0, and the bonds {0, v 0 }, {0, v } 2 are occuied. We denote the neighbours of 0 by e l (l =,..., Ω), so that v 0 = e i and v = e j for some i, j. We eamine the constraints imosed by the event G of (4.42). The event {u C 0 } is satisfied trivially, since u = 0. For the event { C }, we consider searately the cases i = j (i.e., v 0 = v ) and i j (i.e., v 0 v ). If i = j, then { C } requires that {v 0 }, so this 3 is an error term in which (4.44) occurs with j + j 2 + j 3 3 and j 4 = (in more detail, these inequalities imly either that j = 2, in which case j 0 + j + j 2 3 since the sum must be odd, or that j, which imlies that j 2 + j 3 + j 4 3). If i j, then { C } is achieved by the bond {, v 0 } = {0, v 0 }. Thus, we assume henceforth that i j. For the E C0 events, we first note that {v 0 u } occurs, since u = 0, {0, v 0 } is occuied, and 0 C C0 0. Also, {v }2 occurs, since = 0, {0, v } 2 is occuied, and 0 C (when i j). We will argue below that the NP condition in each E event can be neglected, u to an error term. Assuming this, this case contributes 2 + O(Ω 3 ) = Ω(Ω ) 4 + O(Ω 3 ). (4.49) (0,v 0 ) (0,v ): v v 0 6

17 If the NP condition is violated for E (v 0, u ; C 0 ) = E (v 0, 0; C 0 ), then the bond (v 0, 0) must be ivotal for the level- connection from v 0 to 0, and moreover v 0 C 0 = C (0,v 0) 0 (0) must occur. The latter gives an additional factor τ (3) (v 0 ) O(Ω 3/2 ), and hence this contributes to an error term. If the NP condition is violated for E (v, = 0; C ) 2, then the bond (v, 0) 2 must be ivotal for the level-2 connection from v to 0, and also v C = C (0,v ) (v 0 ) must occur. The latter gives an additional factor τ (2) (v v 0 ) O(Ω ), and hence this contributes to an error term. Combining (4.48) (4.49), we have which is (2.0). ˆΠ (2) = Ω 3 + Ω(Ω ) 4 + O(Ω 3 ), (4.50) 5 Conclusions We have used the lace eansion to rove that c (G) = Ω + Ω Ω 3 + O(Ω 4 ) for G = Z n and G = Q n. This etends by two terms the result c (Q n ) = n + O(n 2 ) of [9], and gives a simlified roof of a result of [6, 7] for Z n. Our roof is essentially mechanical, and with sufficient labour could be directly etended to comute higher coefficients. In articular, it would be interesting to comute the coefficient of Ω 4, which we eect will be different for Z n and Q n. We eect that our method can also be alied to other finite grahs for which the lace eansion has been roved to converge in [9]. A secific eamle is the Hamming cube, which has verte set {0,,..., s} n with s fied, and edge set consisting of airs of vertices which differ in eactly one comonent. For s =, the Hamming cube is the n-cube. For s 2, the Hamming cube contains cycles of length 3 (in contrast to Z n and Q n ), and it would be interesting to study their effect on the eansion coefficients. Acknowledgements We thank Christian Borgs, Jennifer Chayes and Joel Sencer for many stimulating discussions related to this work. The work of RvdH was suorted in art by Netherlands Organisation for Scientific Research (NWO), and was carried out in art at Delft University of Technology, at the University of British Columbia, and at Microsoft Research. The work of GS was suorted in art by NSERC of Canada, by a Senior Visiting Fellowshi at the Isaac Newton Institute funded by EPSRC Grant N0976, by EURANDOM, and by the Thomas Stieltjes Institute. References [] M. Aizenman and D.J. Barsky. Sharness of the hase transition in ercolation models. Commun. Math. Phys., 08: , (987). [2] M. Aizenman and C.M. Newman. Tree grah inequalities and critical behavior in ercolation models. J. Stat. Phys., 36:07 43, (984). 7

18 [3] M. Ajtai, J. Komlós, and E. Szemerédi. Largest random comonent of a k-cube. Combinatorica, 2: 7, (982). [4] N. Alon, I. Benjamini, and A. Stacey. Percolation on finite grahs and isoerimetric inequalities. To aear in Ann. Probab. [5] N. Alon and J.H. Sencer. The Probabilistic Method. Wiley, New York, 2nd edition, (2000). [6] B. Bollobás and Y. Kohayakawa. Percolation in high dimensions. Euro. J. Combinatorics, 5:3 25, (994). [7] B. Bollobás, Y. Kohayakawa, and T. Luczak. The evolution of random subgrahs of the cube. Random Struct. Alg., 3:55 90, (992). [8] C. Borgs, J.T. Chayes, R. van der Hofstad, G. Slade, and J. Sencer. Random subgrahs of finite grahs: I. The scaling window under the triangle condition. Prerint, (2003). [9] C. Borgs, J.T. Chayes, R. van der Hofstad, G. Slade, and J. Sencer. Random subgrahs of finite grahs: II. The lace eansion and the triangle condition. To aear in Ann. Probab. [0] C. Borgs, J.T. Chayes, R. van der Hofstad, G. Slade, and J. Sencer. Random subgrahs of finite grahs: III. The hase transition for the n-cube. Prerint, (2003). [] J.T. Co and R. Durrett. Oriented ercolation in dimensions d 4: bounds and asymtotic formulas. Math. Proc. Cambridge Philos. Soc., 93:5 62, (983). [2] D.S. Gaunt and H. Ruskin. Bond ercolation rocesses in d dimensions. J. Phys. A: Math. Gen., : , (978). [3] D.M. Gordon. Percolation in high dimensions. J. London Math. Soc. (2), 44: , (99). [4] G. Grimmett. Percolation. Sringer, Berlin, 2nd edition, (999). [5] T. Hara and G. Slade. Mean-field critical behaviour for ercolation in high dimensions. Commun. Math. Phys., 28:333 39, (990). [6] T. Hara and G. Slade. Unublished aendi to [7]. Available as aer at htt:// arc. (993). [7] T. Hara and G. Slade. The self-avoiding-walk and ercolation critical oints in high dimensions. Combin. Probab. Comut., 4:97 25, (995). [8] R. van der Hofstad and G. Slade. Asymtotic eansions in n for ercolation critical values on the n-cube and Z n. Prerint, (2003). [9] H. Kesten. Asymtotics in high dimensions for ercolation. In G.R. Grimmett and D.J.A. Welsh, editors, Disorder in Physical Systems. Clarendon Press, Oford, (990). [20] M.V. Menshikov. Coincidence of critical oints in ercolation roblems. Soviet Mathematics, Doklady, 33: , (986). 8