Biased periodic Aztec diamond and an elliptic curve

Area of Science:

• Combinatorics
• Statistical Mechanics
• Algebraic Geometry

Background:

• Domino tilings are fundamental in statistical mechanics and combinatorics.
• The Aztec diamond model is a well-studied system with rich mathematical properties.
• Understanding correlations in disordered systems is a key challenge.

Purpose of the Study:

• To develop a novel mathematical framework for biased random domino tilings.
• To derive a double integral formula for the correlation kernel.
• To analyze local correlations in disordered regions of the tiling.

Main Methods:

• Associating a linear flow on an elliptic curve to the tiling model.
• Deriving a double integral formula for the correlation kernel using this flow.
• Employing saddle point analysis for periodic flow cases.
• Investigating a specific case with a period-six flow.

Main Results:

• A double integral formula for the correlation kernel is established.
• Local correlations in the smooth disordered region are computed.
• The boundary of the rough disordered region is identified as an algebraic curve of degree eight for a specific case.

Conclusions:

• The developed framework provides new insights into random domino tilings.
• The connection to elliptic curves offers a powerful tool for analysis.
• The study reveals specific geometric properties of disordered regions in these tilings.