多米诺洗牌高度过程及其流体动力学极限
The domino shuffling height process and its hydrodynamic limit
摘要：
著名的多米诺洗牌算法被发明来生成阿兹特克钻石的多米诺骨牌。利用多米诺高度函数，我们将多米诺洗牌过程看作是平面上离散时间的随机高度过程。推导了任意连续剖面的流体动力学极限为Hamilton-Jacobi方程$u_t+H(u_x)=0$的唯一黏性解，其中$H$的Hessian行列式处处为负.证明涉及离散过程的插值和演化的极限半群的分析。为了确定极限，我们使用二聚体模型的理论以及Hamilton-Jacobi方程。我们的结果是$d>1$中的第一个例子，其中这样一个完整的流体动力学极限与非凸哈密顿量可以得到一个离散系统。我们还定义了更一般的周期性二聚体模型的洗牌高度过程，我们希望类似的结果。
英文摘要：
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 procedure 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 $u_t+H(u_x)=0$, where the determinant of the Hessian of $H$ is negative everywhere. The proof involves interpolation of the discrete process and analysis of 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. It seems that our result is the first example in $d>1$ 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.
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