Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Sorbonne University, 4 Place Jussieu, 75005, Paris, France.
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM, 87501, USA.
Nat Commun. 2023 Feb 4;14(1):618. doi: 10.1038/s41467-023-36233-5.
The territory explored by a random walk is a key property that may be quantified by the number of distinct sites that the random walk visits up to a given time. We introduce a more fundamental quantity, the time τ required by a random walk to find a site that it never visited previously when the walk has already visited n distinct sites, which encompasses the full dynamics about the visitation statistics. To study it, we develop a theoretical approach that relies on a mapping with a trapping problem, in which the spatial distribution of traps is continuously updated by the random walk itself. Despite the geometrical complexity of the territory explored by a random walk, the distribution of the τ can be accounted for by simple analytical expressions. Processes as varied as regular diffusion, anomalous diffusion, and diffusion in disordered media and fractals, fall into the same universality classes.
随机游走所探索的区域是一个关键性质，可以通过随机游走在给定时间内访问的不同位置的数量来量化。我们引入了一个更基本的量，即当随机游走已经访问了 n 个不同的位置时，找到一个它以前从未访问过的位置所需的时间 τ，这包含了关于访问统计数据的全部动态信息。为了研究它，我们开发了一种理论方法，该方法依赖于与捕获问题的映射，其中陷阱的空间分布由随机游走本身不断更新。尽管随机游走所探索的区域的几何形状很复杂，但 τ 的分布可以用简单的解析表达式来描述。各种过程，如规则扩散、异常扩散、无序介质和分形中的扩散，都属于相同的普遍性类别。
