Expansion of Percolation Critical Points for Hamming Graphs. (5th January 2020) - Federico, Lorenzo; Van Der Hofstad, Remco; Den Hollander, Frank; Hulshof, Tim
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 N fixed and n → ∞, p_c^(d) = 1 m + 2d^2 - 1 2(d - 1)^21 m^2 + 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 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.