并矢拼接的相变
The Phase Transition for Dyadic Tilings
摘要：
一个n阶的并元瓦片是通过水平或垂直切割从单位正方形通过n个连续二等分获得的任何矩形。令每个n阶的二元图块以概率p可用，独立于其他图块。我们证明，对于p足够接近1，存在一组成对不相交的可用瓦片，其并集是单位平方，当n->无穷大时概率趋于1，正如Joel Spencer在1999年所证明的那样。特别地，我们证明了，如果p=7/8，这样的平铺存在的概率至少为1-（3/4）^n。这个证明涉及到一个令人惊讶的微妙的计数参数，用于防止平铺的不可用的瓷砖集。
英文摘要：
A dyadic tile of order n is any rectangle obtained from the unit square by n successive bisections by horizontal or vertical cuts. Let each dyadic tile of order n be available with probability p, independently of the others. We prove that for p sufficiently close to 1, there exists a set of pairwise disjoint available tiles whose union is the unit square, with probability tending to 1 as n->infinity, as conjectured by Joel Spencer in 1999. In particular we prove that if p=7/8, such a tiling exists with probability at least 1-(3/4)^n. The proof involves a surprisingly delicate counting argument for sets of unavailable tiles that prevent tiling.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
