Wiener-Hopf factorizations and matrix-valued orthogonal polynomials

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 2-periodic Aztec diamond in terms of Jacobi theta functions related to the spectral curve of the model.