THE PHASE TRANSITION FOR DYADIC TILINGS
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY(2014)
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, independent 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.
更多查看译文
关键词
Dyadic rectangle,tiling,phase transition,percolation,generating function
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络