The famous domino shuffling algorithm was invented to generate the domino tilings of the Aztec Diamond. Using the domino height function, we view the domino shuffling algorithm as a discrete-time random height process on the plane. The hydrodynamic limit from an arbitrary continuous profile is deduced to be the unique viscosity solution of a Hamilton-Jacobi equation, where the determinant of the Hessian of the Hamiltonian is negative everywhere. The proof involves smoothing the discrete process and analyzing the limiting semigroup of the evolution. In order to identify the limit, we use the theories of dimer models as well as Hamilton-Jacobi equations. We also prove the scaling limit of the Aztec Diamond, which implies a limit shape for random domino tilings. It seems that our result is the first example in dimension greater than one where such a full hydrodynamic limit with a nonconvex Hamiltonian can be obtained for a discrete system. We also define the shuffling height process for more general periodic dimer models, where we expect similar results to hold. We also include a discussion of a 1-dimensional exclusion process closely related to the domino shuffling algorithm, which was first invented to prove the Arctic Circle Theorem.
Zhang, Xufan,
"The Domino Shuffling Height Process and Its Hydrodynamic Limit"
(2019).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/zrs9-z066
