Logarithmic fluctuations for internal DLA
Logarithmic fluctuations for internal DLA
复制标题
内部 DLA 的对数波动
DOI：
10.1090/s0894-0347-2011-00716-9
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
S. Sheffield
中科院分区：
文献类型：
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作者：
David Jerison;Lionel Levine;S. Sheffield
让从Z^2的原点开始的n个粒子进行简单的随机行走，直到到达没有其他粒子的位置。Lawler，Bramson和Griffeath证明了所得到的n个被占用站点的随机集A(N)(概率很高)接近半径为r=\Sqrt{n/\pi}的圆盘B_r。我们证明了A(N)与圆盘之间的偏差在半径上至多是对数的：即存在一个绝对常数C，使得下述条件以概率1成立：对于所有足够大的r，B_{r-C\logr}\子集A(\pi r^2)\子集B_{r+C\logr}。
Let each of n particles starting at the origin in Z^2 perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resulting random set A(n) of n occupied sites is (with high probability) close to a disk B_r of radius r=\sqrt{n/\pi}. We show that the discrepancy between A(n) and the disk is at most logarithmic in the radius: i.e., there is an absolute constant C such that the following holds with probability one: B_{r - C \log r} \subset A(\pi r^2) \subset B_{r+ C \log r} for all sufficiently large r.