Biased periodic Aztec diamond and an elliptic curve.

We study random domino tilings of the Aztec diamond with a biased periodic weight function and associate a linear flow on an elliptic curve to this model. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of this flow. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel.