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arXiv · 2410.17138

Arctic curves of periodic dimer models and generalized discriminants

Mateusz Piorkowski
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Abstract

We compute the algebraic equation for arctic curves of the Aztec diamond with a doubly (quasi-)periodic weight structure and obtain similar results for certain models of the hexagon. In particular, we determine the algebraic degree of such curves as a function of the number of frozen and smooth (or gaseous) regions. The key to our result is the construction of a discriminant for meromorphic differentials on a higher genus Riemann surface. This construction works analogously for meromorphic sections of arbitrary holomorphic line bundles. In the genus $g = 0$ case this notion reduces to the usual discriminant of a polynomial.

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Mateusz Piorkowski. 2024-10-22. Arctic curves of periodic dimer models and generalized discriminants. https://arxiv.org/abs/2410.17138

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