Laboratoire de Physique Théorique et Modèles Statistiques (UMR 8626 du CNRS), Université Paris-Sud, Bâtiment 100, 91405 Orsay Cedex, France.
Phys Rev Lett. 2013 May 31;110(22):220602. doi: 10.1103/PhysRevLett.110.220602. Epub 2013 May 29.
We study the number of distinct sites S(N)(t) and common sites W(N)(t) visited by N independent one dimensional random walkers, all starting at the origin, after t time steps. We show that these two random variables can be mapped onto extreme value quantities associated with N independent random walkers. Using this mapping, we compute exactly their probability distributions P(N)(d)(S,t) and P(N)(c)(W,t) for any value of N in the limit of large time t, where the random walkers can be described by Brownian motions. In the large N limit one finds that S(N)(t)/√t∝2√(log N)+s/(2√(log N)) and W(N)(t)/√t∝w/N where s and w are random variables whose probability density functions are computed exactly and are found to be nontrivial. We verify our results through direct numerical simulations.
我们研究了在 t 时间步后，N 个独立的一维随机行走者从原点出发访问的不同站点数 S(N)(t)和公共站点数 W(N)(t)。我们表明，这两个随机变量可以映射到与 N 个独立随机行走者相关的极值量上。使用这种映射，我们精确地计算了它们的概率分布 P(N)(d)(S,t)和 P(N)(c)(W,t)，对于任意 N 值，在大时间 t 的极限下，随机行走者可以用布朗运动来描述。在大 N 极限下，我们发现 S(N)(t)/√t∝2√(log N)+s/(2√(log N))和 W(N)(t)/√t∝w/N，其中 s 和 w 是随机变量，其概率密度函数被精确计算，并且被发现是非平凡的。我们通过直接数值模拟验证了我们的结果。
