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

Alexei Borodin; Maurice Duits

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arXiv · 2203.11885

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

Alexei Borodin·Maurice Duits

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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.

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BibTeX RIS

Alexei Borodin, Maurice Duits. 2022-03-22. Biased $2 \times 2$ periodic Aztec diamond and an elliptic curve. https://arxiv.org/abs/2203.11885

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