$\Z^d$中抛硬币指数尾的不变匹配
Invariant matchings of exponential tail on coin flips in $\Z^d$
摘要：
考虑$\Z^d$上的Bernoulli（1/2）渗流，并定义开顶点和闭顶点之间的完美匹配，其方式是配置的确定性等变函数。我们希望找到这样的匹配规则，使得原点对的距离大于$r$的概率尽可能快地衰减。对于二维情形，我们给出了一个最优的衰减匹配$cr^{1/2}$.对于至少3维，我们给出了一个具有指数尾的匹配规则。这大大改善了以前的边界。构造有两个主要部分：首先，我们定义了一个序列的粗和粗分区的$\Z^d$在等变的方式，这样的高概率的细胞的不动点是像一个立方体，它的标签是独立同分布。然后，我们定义一个固定的有限单元格的匹配，当我们重复应用于连续分区的单元格时，它会稳定下来。我们的方法也工作的情况下，当一个人想要匹配点的两个泊松过程，他们可能会被应用到分配问题。
英文摘要：
Consider Bernoulli(1/2) percolation on $\Z^d$, and define a perfect matching between open and closed vertices in a way that is a deterministic equivariant function of the configuration. We want to find such matching rules that make the probability that the pair of the origin is at distance greater than $r$ decay as fast as possible. For two dimensions, we give a matching of decay $cr^{1/2}$, which is optimal. For dimension at least 3 we give a matching rule that has an exponential tail. This substantially improves previous bounds. The construction has two major parts: first we define a sequence of coarser and coarser partitions of $\Z^d$ in an equivariant way, such that with high probability the cell of a fixed point is like a cube, and the labels in it are i.i.d. Then we define a matching for a fixed finite cell, which stabilizes as we repeatedly apply it for the cells of the consecutive partitions. Our methods also work in the case when one wants to match points of two Poisson processes, and they may be applied to allocation questions.
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