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 m= d(n- 1) be the degree and V = n(d) 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 is an element of N fixed and n -> infinity, p(c)((d)) = 1/m + 2d(2) - 1/2(d - 1)(2) 1/m(2) + O(m(-3)) + O(m(-1)V(-1/3)), which extends the asymptotics found in [10] by one order. The term O(m(-1)V(-1/3)) is the width of the critical window. For d = 4, 5, 6 we have m(-3) = O(m(-1)V(-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 d = 2, 3, 4. 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 Erdos-Renyi random graph.