Parity-dependent sublattice structure of local $2\times2$ correlations in domino tilings of the Aztec diamond
We study local $2\times2$ correlations in uniformly random domino tilings of the Aztec diamond $\mathrm{AZ}(n)$, a dimer model with frozen corners and a liquid region bounded by the Arctic circle, and compare them with the square chessboard. For every elementary square the probabilities $P_H$ and $P_V$ of the two coverings by a pair of parallel dominoes satisfy $P_H=P_V=P_1P_2+P_3P_4$, where $P_1,\dots,P_4$ are the probabilities of the four surrounding edges. This known identity, a case of Kuo's graphical condensation, reduces the spatial profile of $P_H$ to single-edge probabilities, which we compute exactly from the inverse Kasteleyn matrix for all squares of $\mathrm{AZ}(n)$. The squares form two interleaved sublattices, distinguished by the color of their lower-left cell. The parity of $n$ decides which sublattice dominates near the center: the one containing the central square for odd $n$, and the other one for even $n$. This sign rule holds for the averaged sublattice contrast for all $4\le n\le22$. It also fixes the location of the global maximum of $P_H$, which lies at or next to the center for odd $n$ and several lattice spacings away for even $n$. Within the central sublattice, the preferred square follows a mod-4 pattern that matches the exactly pinned mean height of the center. On the $L\times L$ chessboard the same rule holds with $n$ replaced by the half-width $L/2$, but the contrast is two to six times weaker, the central edge probability never equals $1/4$, and the maxima of $P_H$ are pinned to the corners. The exact $1/4$ values and the interior, parity-selected maxima are therefore features of the Aztec geometry.