Expansion of percolation critical points for hamming graphs

The Hamming graph H(d, n) is the Cartesian product of d complete graphs on n vertices. Let be the degree and be the number of vertices of H(d, n). Let pc(d) be the critical point for bond percolation on H(d, n). We show that, for d ∈ ℕ fixed and n → ∞, pc(d) = 1/m + 2d2 - 1/2(d -1)2 1/m2 + O (m-3)+O (m-1V-1/3), which extends the asymptotics found in [10] by one order. The term O(m-1V-1/3) is the width of the critical window. For d=4,5,6 we have & m-3 = O(m-1V-1/3), and so the above formula represents the full asymptotic expansion of pc(d). In [16] we show that this formula is a crucial ingredient in the study of critical bond percolation on H(d, n) for. The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erdös-Rényi random graph.