Abstract
We analyze domino tilings of the two-periodic Aztec diamond by means of matrix valued orthogonal polynomials that we obtain from a reformulation of the Aztec diamond as a non-intersecting path model with periodic transition matrices. In a more general framework we express the correlation kernel for the underlying determinantal point process as a double contour integral that contains the reproducing kernel of matrix valued orthogonal polynomials. We use the Riemann-Hilbert problem to simplify this formula for the case of the two-periodic Aztec diamond. In the large size limit we recover the three phases of the model known as solid, liquid and gas. We describe fine asymptotics for the gas phase and at the cusp points of the liquid-gas boundary, thereby complementing and extending results of Chhita and Johansson.
Highlights
We study domino tilings of the Aztec diamond with a two periodic weighting
This model falls into a class of models for which existing techniques for studying fine asymptotics are not adequate and only recently first important progress has been made [9, 20]
We introduce a new approach based on matrix valued orthogonal polynomials that allows us to compute the determinantal correlations at finite size and their asymptotics as the size of the diamond gets large in a rather orderly way
Summary
We study domino tilings of the Aztec diamond with a two periodic weighting. This model falls into a class of models for which existing techniques for studying fine asymptotics are not adequate and only recently first important progress has been made [9, 20]. For a general class of discrete non-intersecting paths with p-periodic transition matrices (which includes p-periodic weightings for domino tilings of the Aztec diamond and p-periodic weightings for lozenge tilings of the hexagon), we show in Section 4 how the correlation kernel can be written as a double integral formula involving matrix valued polynomials that satisfy a non-hermitian orthogonality. We believe that this general setup has a high potential for a rigorous asymptotic analysis. The model of the two periodic Aztec diamond is explained and the main results are summarized
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