How long does it take for Internal DLA to forget its initial profile?
How long does it take for Internal DLA to forget its initial profile?
复制标题
内部 DLA 需要多长时间才能忘记其初始配置文件？
DOI：
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发表时间：
2018
影响因子：
2
通讯作者：
Vittoria Silvestri
中科院分区：
文献类型：
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作者：
Lionel Levine;Vittoria Silvestri
内部DLA是运动界面的离散模型。在圆柱图$${mathbb{Z}}_N imes{{mathbb{Z}}$$ZN×Z上，一个粒子均匀地从$${mathbb{Z}}_N imes{0}$$ZN×{0}开始，在柱面上执行简单的随机行走，直到到达它永远占据的$${mathbb{Z}}_N imes{{mathbb{Z}_{ge 0}$$ZN×Z≥0中的一个空置位置。该操作定义了圆柱体的子集上的马尔可夫链。我们首先证明了一个典型的子集是最多具有对数起伏的矩形。利用这一点，我们证明了通过增加阶数$$N^2log N$$N2logN粒子，从不同的典型子集开始的两个内部DLA链可以高概率地耦合。对于下界，我们证明了至少需要阶数为$$N^2$$N_2的粒子来忘记该过程是从两个独立的典型子集中的哪一个开始的。
Internal DLA is a discrete model of a moving interface. On the cylinder graph $${{mathbb {Z}}}_N imes {{mathbb {Z}}}$$ZN×Z, a particle starts uniformly on $${{mathbb {Z}}}_N imes {0}$$ZN×{0} and performs simple random walk on the cylinder until reaching an unoccupied site in $${{mathbb {Z}}}_N imes {{mathbb {Z}}}_{ge 0}$$ZN×Z≥0, which it occupies forever. This operation defines a Markov chain on subsets of the cylinder. We first show that a typical subset is rectangular with at most logarithmic fluctuations. We use this to prove that two Internal DLA chains started from different typical subsets can be coupled with high probability by adding order $$N^2 log N$$N2logN particles. For a lower bound, we show that at least order $$N^2$$N2 particles are required to forget which of two independent typical subsets the process started from.