Asymptotic expansions in n−1 for percolation critical values on the n-cube and Z

We use the lace expansion to prove that the critical values for nearest-neighbour bond percolation on the n-cube {0, 1}n and on Zn have asymptotic expansions, with rational coefficients, to all orders in powers of n−1. 1 Main result We consider bond percolation on Z with edge set consisting of pairs {x, y} of vertices in Z with ‖x−y‖1 = 1, where ‖w‖1 = nj=1 |wj| for w ∈ Z. Bonds (edges) are independently occupied with probability p and vacant with probability 1− p. We also consider bond percolation on the n-cube Qn, which has vertex set {0, 1}n and edge set consisting of pairs {x, y} of vertices in {0, 1}n with ‖x− y‖1 = 1, where we regard Qn as an additive group with addition component-wise modulo 2. Again bonds are independently occupied with probability p and vacant with probability 1−p. We write G in place of Qn and Z when we wish to refer to both models simultaneously. We write Ω for the degree of G, so that Ω = 2n for Z and Ω = n for Qn. For the case of Z, the critical value is defined by pc(Z) = inf{p : ∃ an infinite connected cluster of occupied bonds a.s.}. (1.1) 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 of C(x), and let χ(p) = Ep|C(0)| denote the expected cluster size of the origin. Results of [1, 20] imply that pc(Z) = sup{p : χ(p) < ∞}. (1.2) is an equivalent definition of the critical value. ∗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]