Arno Kuijlaars (KU Leuven)
The two-periodic Aztec diamond and matrix valued orthogonal polynomials
Uniform domino tilings of the Aztec diamond have the arctic circle
phenomenon: near the corners the pattern is fixed and only one type of domino
appears, while in the middle there is disorder and all types appear.
The transition is sharp with fluctuations described
by the Tracy-Widom distributions.
In the two-periodic Aztec diamond the dominos have a two-periodic weighting
and this creates a new phase in the large size limit, where correlations
decay at an exponential rate. In recent work with Maurice Duits (KTH Stockholm)
we analyze this model with the help of matrix valued orthogonal
polynomials. We obtain a remarkably simple double contour integral
formula for the correlation kernel that we can analyze in the limit to
recover the three phases of the model and the fluctuations near the