The Domino Shuffling Height Process and Its Hydrodynamic Limit
The Domino Shuffling Height Process and Its Hydrodynamic Limit
复制标题
多米诺骨牌洗牌高度过程及其水动力极限
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
Xufan Zhang
中科院分区：
文献类型：
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作者：
Xufan Zhang
著名的多米诺洗牌算法是为了生成阿兹特克钻石的多米诺骨牌拼图而发明的。利用多米诺骨牌高度函数，我们将多米诺骨牌洗牌过程看作是平面上离散时间的随机高度过程。推导出任意连续剖面的流体动力极限是Hamilton-Jacobi方程$u_t+H(U_X)=0$的唯一粘性解，其中$H$的Hessian行列式处处为负。证明包括离散过程的插值法和演化极限半群的分析。为了确定极限，我们使用了二聚体模型理论和哈密顿-雅可比方程。
我们的结果似乎是$d>；1中的第一个例子，其中对于离散系统，可以获得具有非凸哈密顿量的完全流体动力学极限。我们还定义了更一般的周期二聚体模型的洗牌高度过程，我们预计类似的结果也会成立。
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.