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

TL;DR

This paper investigates the placement probabilities of domino tilings in the Aztec diamond, revealing a consistent 1/4-phenomenon and deriving explicit counting formulas for tilings with holes.

Contribution

It proves a universal 1/4-phenomenon for domino placement probabilities and provides explicit counting formulas for tilings with arbitrary holes, advancing combinatorial enumeration methods.

Findings

Placement probability is always 1/4 plus a location-dependent rational function.

Derived explicit formulas for tilings with 2x2 holes at arbitrary positions.

Simplified counting formulas compared to previous methods.

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.