平铺图上沙堆的截断
Cut-off for sandpiles on tiling graphs
摘要：
沙堆动力学被认为是从周期性的平面和空间平铺通过分配一个不断增长的块的平铺环或开放的边界条件构建的图形。给出了求镶嵌格林函数的一般方法，并证明了一般条件下的全变差截止现象。结果表明，边界条件不影响混合时间的平面平铺，也不改变渐近混合时间的立方晶格$\zed^d$所有足够大的$d$。在配套文件中，计算方法被用来证明，混合时间被改变的$\Dfour$晶格在4维。
英文摘要：
Sandpile dynamics are considered on graphs constructed from periodic plane and space tilings by assigning a growing piece of the tiling either torus or open boundary conditions. A general method of obtaining the Green's function of the tiling is given, and a total variation cut-off phenomenon is demonstrated under general conditions. It is shown that the boundary condition does not affect the mixing time for planar tilings, nor does it change the asymptotic mixing time for the cubic lattice $\zed^d$ for all sufficiently large $d$. In a companion paper, computational methods are used to demonstrate that the mixing time is altered for the $\Dfour$ lattice in dimension 4.
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