Arctic curves of periodic dimer models and generalized discriminants

TL;DR

This paper derives algebraic equations for arctic curves in periodic dimer models, using discriminants on Riemann surfaces, and determines their degrees based on the model's frozen and gaseous regions.

Contribution

It introduces a novel discriminant construction for meromorphic differentials on higher genus Riemann surfaces, generalizing polynomial discriminants to complex algebraic curves.

Findings

Algebraic equations for arctic curves in periodic dimer models are obtained.

The algebraic degree of these curves depends on the number of frozen and gaseous regions.

Discriminant construction extends to higher genus Riemann surfaces, generalizing polynomial discriminants.

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.