Biased $2 \times 2$ periodic Aztec diamond and an elliptic curve

TL;DR

This paper investigates biased periodic domino tilings of the Aztec diamond, linking them to elliptic curves, and derives explicit formulas for local correlations, especially in special periodic cases.

Contribution

It introduces a novel connection between biased tilings and elliptic curves, providing explicit integral formulas and analyzing special periodic cases.

Findings

Derived a double integral formula for the correlation kernel.

Identified conditions under which the flow is periodic and analyzed local correlations.

Explicitly computed the boundary of the rough disordered region as an algebraic curve.

Abstract

We study a biased $2\times 2$ periodic random domino tilings of the Aztec diamond and associate a linear flow on an elliptic curve to this model. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of this flow. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel. In these cases we compute the local correlations in the smooth disordered (or gaseous) region. The special example in which the flow has period six is worked out in more detail, and we show that in that case the boundary of the rough disordered region is an algebraic curve of degree eight.