Expansion of Percolation Critical Points for Hamming Graphs

Abstract

AbstractThe Hamming graphH(d,n) is the Cartesian product ofdcomplete graphs onnvertices. Let${m=d(n-1)}$be the degree and$V = n^d$be the number of vertices ofH(d,n). Let$p_c^{(d)}$be the critical point for bond percolation onH(d,n). We show that, for$d \in \mathbb{N}$fixed and$n \to \infty$,$$p_c^{(d)} = {1 \over m} + {{2{d^2} - 1} \over {2{{(d - 1)}^2}}}{1 \over {{m^2}}} + O({m^{ - 3}}) + O({m^{ - 1}}{V^{ - 1/3}}),$$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 onH(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 Erdös–Rényi random graph.