Kiumi Chusei, Konno Norio, Tamura Shunya
Graduate School of Science and Engineering, Yokohama National University, Hodogaya, Yokohama 240-8501, Japan.
Department of Applied Mathematics, Faculty of Engineering, Yokohama National University, Hodogaya, Yokohama 240-8501, Japan.
Entropy (Basel). 2022 Apr 21;24(5):584. doi: 10.3390/e24050584.
The analysis of the return probability is one of the most essential and fundamental topics in the study of classical random walks. In this paper, we study the return probability of quantum and correlated random walks in the one-dimensional integer lattice by the path counting method. We show that the return probability of both quantum and correlated random walks can be expressed in terms of the Legendre polynomial. Moreover, the generating function of the return probability can be written in terms of elliptic integrals of the first and second kinds for the quantum walk.
返回概率的分析是经典随机游走研究中最基本和重要的主题之一。在本文中，我们通过路径计数方法研究了一维整数格上量子和相关随机游走的返回概率。我们表明，量子和相关随机游走的返回概率都可以用勒让德多项式表示。此外，对于量子游走，返回概率的生成函数可以用第一类和第二类椭圆积分表示。
