Expansion of Percolation Critical Points for Hamming Graphs

Source abstract

Abstract 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 $d \in \mathbb{N}$ 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 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 Erdös–Rényi random graph.