Vacant set of random interlacements and percolation
成果类型：
Article
署名作者：
Sznitman, Alain-Sol
刊物名称：
ANNALS OF MATHEMATICS
ISSN/ISSBN：
0003-486X
发表日期：
2010
页码：
2039-2087
关键词：
摘要：
We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z(d), d >= 3. A nonnegative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the microscopic structure in the bulk, which arises when considering the disconnection time of a discrete cylinder (Z/NZ)(d-1) x Z by simple random walk, or the set of points visited by simple random walk on the discrete torus (Z/NZ)(d) at times of order uN(d). In particular we study the percolative properties of the vacant set left by the interlacement at level u, which is an infinite connected translation invariant random subset of Z d. We introduce a critical value u(*) such that the vacant set percolates for u < u(*) and does not percolate for u > u(*). Our main results show that u(*) is finite when d >= 3 and strictly positive when d >= 7.