Séminaire Lotharingien de Combinatoire, 95B.50 (2026), 12 pp.
Marcus Schönfelder
The 1/4-Phenomenon of Placement Probabilities in the Aztec Diamond
Abstract.
We consider domino tilings of the Aztec diamond. Using the Domino
Shuffling algorithm introduced by Elkies, Kuperberg, Larsen, and
Propp (1992), 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 × 2-square holes at arbitrary positions.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
The following versions are available: