For a symmetric subclass of 3x3-periodic hexagon tilings, the associated matrix valued orthogonal polynomials admit a full asymptotic expansion built from an explicit balayage equilibrium measure on a genus one spectral curve.
Arctic curves of periodic dimer models and generalized discriminants
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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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Matrix valued orthogonal polynomials arising from hexagon tilings with 3x3-periodic weightings
For a symmetric subclass of 3x3-periodic hexagon tilings, the associated matrix valued orthogonal polynomials admit a full asymptotic expansion built from an explicit balayage equilibrium measure on a genus one spectral curve.