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 p_c^(d) be the critical point for bond percolation on H(d, n). We show that, for d ∈ ℕ fixed and n → ∞, p_c^(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 p_c^(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.