Selecta Mathematica· 2026Q1
Arctic curves of periodic dimer models and generalized discriminants
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- 2026year
Short summary
A new algebraic equation for arctic curves of the Aztec diamond and hexagon models with doubly periodic weights is derived, with its degree determined by frozen and smooth regions.
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Key points
- Derived the algebraic equation for arctic curves of the Aztec diamond and hexagon models with doubly periodic weights.
- Determined the algebraic degree of these curves based on the number of frozen and smooth regions.
- Constructed a discriminant for meromorphic differentials on higher genus Riemann surfaces.
- The construction generalizes the standard polynomial discriminant for genus 0 surfaces.
AI-generated from the title and abstract; the full text is not read.
Abstract
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 $$g = 0$$ g = 0 case this notion reduces to the usual discriminant of a polynomial.
The authors' abstract, as published at the source. Selecta Mathematica, 2026 · DOI ↗
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