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The 2-dimensional Hamming graph H(2,n) consists of the $n^2$ vertices$(i,j)$, $1\leq i,j\leq n$, two vertices being adjacent when they share acommon coordinate. We examine random subgraphs of H(2,n) in percolation withedge probability $p$, so that the average degree $2(n-1)p=1+\epsilon$. Previouswork by van der Hofstad and Luczak had shown that in the barely supercriticalregion $n^{-2/3}\ln^{1/3}n\ll \epsilon \ll 1$ the largest component has size$\sim 2\epsilon n$. Here we show that the second largest component has sizeclose to $\epsilon^{-2}$, so that the dominant component has emerged. Thisresult also suggests that a {\it discrete duality principle} might hold,whereby, after removing the largest connected component in the supercriticalregime, the remaining random subgraphs behave as in the subcritical regime.

The second largest component in the supercritical 2D Hamming graph