论文标题
与固定，随机和收缩的步骤随机步行中的相似性和自相似性
Similarity and self-similarity in random walk with fixed, random and shrinking steps
论文作者
论文摘要
在本文中，我们首先对随机步行（RW）问题进行全面描述，该问题的重点是自相似性，动态缩放及其与扩散现象的联系。我们工作的主要目标之一是检查RW问题在步骤尺寸的各种不同选择下的鲁棒性。我们表明，具有随机步长或均匀缩小步长的RW与固定步长的RW完全相同。 Krapivsky和Redner于2004年表明，带有几何缩小的步骤尺寸的RW，因此$ n $ th步骤的大小由$ s_n =λ^n $带有固定的$λ<1 $值，具有一些有趣的功能，这些功能与固定步骤尺寸的RW不同。由此激励，我们调查了$λ$不是固定的数字，而是取决于步骤号$ n $？为此，我们首先生成$ n $ $ t = n $的$ n $随机数，然后以降序排列，以使$ n $ th the步骤的大小为$λ_n^n $。我们已经在数值和分析上表明，$λ_n=（1-n/n）$，均方根位移的增加为$ t^{1/4} $，与RW问题上的所有已知结果不同。
In this article, we first give a comprehensive description of random walk (RW) problem focusing on self-similarity, dynamic scaling and its connection to diffusion phenomena. One of the main goals of our work is to check how robust the RW problem is under various different choices of the step size. We show that RW with random step size or uniformly shrinking step size is exactly the same as for RW with fixed step size. Krapivsky and Redner in 2004 showed that RW with geometric shrinking step size, such that the size of the $n$th step is given by $S_n=λ^n$ with a fixed $λ<1$ value, exhibits some interesting features which are different from the RW with fixed step size. Motivated by this, we investigate what if $λ$ is not a fixed number rather it depends on the step number $n$? To this end, we first generate $N$ random numbers for RW of $t=N$ which are then arranged in a descending order so that the size of the $n$th step is $λ_n^n$. We have shown, both numerically and analytically, that $λ_n=(1-n/N)$, the root mean square displacement increases as $t^{1/4}$ which are different from all the known results on RW problems.
