ABSTRACT:
Let $R$ be a region in the plane consisting of a union of unit squares which align with the square grid $Z^{^2}$. A domino tiling of $R$ is a covering of the region with $1\times 2$- or $2\times 1$-rectangles called domino tiles. Given $R$, enumerative combinatorics asks for the number of domino tilings. Naturally, This number heavily depends on the underlying region $R$.
We consider domino tilings of the Aztec diamond, a well studied model region in this discipline. 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 talk, 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.