On a general many-dimensional excited random walk
On a general many-dimensional excited random walk
复制标题
一般多维激发随机游走
DOI：
10.1214/11-aop678
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发表时间：
2010
影响因子：
2.3
通讯作者：
M. Vachkovskaia
中科院分区：
文献类型：
--
作者：
M. Menshikov;S. Popov;Alejandro F. Ram'irez;M. Vachkovskaia
在本文中，我们研究了一个实质性的推广模型的激发随机游动介绍[电子。可能吧8（2003）86-92]。考虑离散时间随机过程（Xn，n= 0，1，2，.），取值于Zd，d≥2，描述如下：当粒子第一次访问一个站点时，它在给定方向l上有一致正漂移;当粒子在以前访问过的站点时，它有零漂移.假设过程是一致椭圆性的，且过程的跳跃是一致有界的，我们证明了过程在l方向上是弹道的，使得lim infn→∞Xn <$ln>0. 证明这一结果的一个关键因素是对过程在时间n之前访问少于n1/2+α个不同站点的概率的估计，其中α是取决于模型参数的某个正数。这种方法完全避免了使用tan点和特定于受激随机游走的耦合方法。此外，我们应用这种技术，以证明在一个独立同分布的激发随机游动。随机环境满足弹道大数定律和中心极限定理。
In this paper we study a substantial generalization of the model of excited random walk introduced in [Electron. Commun. Probab. 8 (2003) 86–92] by Benjamini and Wilson. We consider a discrete-time stochastic process (Xn,n=0,1,2,…) taking values on Zd, d≥2, described as follows: when the particle visits a site for the first time, it has a uniformly-positive drift in a given direction l; when the particle is at a site which was already visited before, it has zero drift. Assuming uniform ellipticity and that the jumps of the process are uniformly bounded, we prove that the process is ballistic in the direction l so that lim infn→∞Xn⋅ln>0. A key ingredient in the proof of this result is an estimate on the probability that the process visits less than n1/2+α distinct sites by time n, where α is some positive number depending on the parameters of the model. This approach completely avoids the use of tan points and coupling methods specific to the excited random walk. Furthermore, we apply this technique to prove that the excited random walk in an i.i.d. random environment satisfies a ballistic law of large numbers and a central limit theorem.