Title: Expansion of percolation critical points for Hamming graphs,
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 $n \to
\infty$,
\begin{equation*}
p_c^{(d)}= \dfrac{1}{m} + \dfrac{2d^2-1}{2(d-1)^2}\dfrac{1}{m^2}
+ O(m^{-3}) + O(m^{-1}V^{-1/3}),
\end{equation*} which extends the asymptotics found in
\cite{BorChaHofSlaSpe05b} 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 \cite{FedHofHolHul16a} \st{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 Erdős-Rényi random graph.