Parity-dependent sublattice structure of local 2×2 correlations in domino tilings of the Aztec diamond
Abstract
We study local 2×2 correlations in uniformly random domino tilings of the Aztec diamond 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 PH and PV of the two coverings by a pair of parallel dominoes satisfy PH=PV=P1P2+P3P4, where P1,…,P4 are the probabilities of the four surrounding edges. This known identity, a case of Kuo's graphical condensation, reduces the spatial profile of PH to single-edge probabilities, which we compute exactly from the inverse Kasteleyn matrix for all squares of 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≤ n≤22. It also fixes the location of the global maximum of PH, 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× 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 PH are pinned to the corners. The exact 1/4 values and the interior, parity-selected maxima are therefore features of the Aztec geometry.
Related
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BibTeX
@misc{kim2026paritydependent,
title = {{Parity-dependent sublattice structure of local $2\times2$ correlations in domino tilings of the Aztec diamond}},
author = {Chaeyeon Kim and Suhwan Yoo and Jeehoon Choi and Hee Young Kwon and Changyeon Won},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22974},
eprint = {2610.22974},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22974},
}
RIS
TY - UNPB
TI - Parity-dependent sublattice structure of local 2×2 correlations in domino tilings of the Aztec diamond
AU - Chaeyeon Kim
AU - Suhwan Yoo
AU - Jeehoon Choi
AU - Hee Young Kwon
AU - Changyeon Won
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22974
UR - https://arxiv.org/abs/2610.22974
AB - We study local 2×2 correlations in uniformly random domino tilings of the Aztec diamond 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,…,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 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≤ n≤22. 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× 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.
ER -