Random walks with different directions
成果类型：
Article
署名作者：
Herdade, Simo; Vu, Van
署名单位：
Rutgers University System; Rutgers University New Brunswick; Yale University
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-015-0635-7
发表日期：
2016
页码：
1071-1078
关键词：
摘要：
As an extension of Polya's classical result on random walks on the square grids (), we consider a random walk where the steps, while still have unit length, point to different directions. We show that in dimensions at least 4, the returning probability after n steps is at most , which is sharp. The real surprise is in dimensions 2 and 3. In dimension 2, where the traditional grid walk is recurrent, our upper bound is , which is much worse than in higher dimensions. In dimension 3, we prove an upper bound of order . We find a new conjecture concerning incidences between spheres and points in , which, if holds, would improve the bound to , which is consistent to the case. This conjecture resembles Szemer,di-Trotter type results and is of independent interest.