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.
Wiener-Hopf factorizations and matrix-valued orthogonal polynomials
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abstract
We compare two methods for analysing periodic dimer models. These are the matrix-valued orthogonal polynomials approach due to Duits and one of the authors, and the Wiener-Hopf approach due to Berggren and Duits. We establish their equivalence in the special case of the Aztec diamond. Additionally, we provide explicit formulas for the matrix-valued orthogonal polynomials/Wiener-Hopf factors in the case of the $2 \times 2$-periodic Aztec diamond in terms of Jacobi theta functions related to the spectral curve of the model.
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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.