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arXiv · 2512.08377

The 1/4-phenomenon of placement probabilities of tilings in the Aztec diamond

Marcus Schönfelder
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Abstract

We consider domino tilings of the Aztec diamond. Using the Domino Shuffling algorithm introduced by Elkies, Kuperberg, Larsen, and Propp in arXiv:math/9201305, we are able to generate domino tilings uniformly at random. In this paper, we investigate the probability of finding a domino at a specific position in such a random tiling. We prove that this placement probability is always equal to $1/4$ plus a rational function, whose shape depends on the location of the domino, multiplied by a position-independent factor that involves only the size of the diamond. This result leads to significantly more compact explicit counting formulas compared to previous findings. As a direct application, we derive explicit counting formulas for the domino tilings of Aztec diamonds with $2\times 2$-square holes at arbitrary positions.

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Marcus Schönfelder. 2025-12-09. The 1/4-phenomenon of placement probabilities of tilings in the Aztec diamond. https://arxiv.org/abs/2512.08377

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