摘要:随机行走在很多领域里都有重要的应用，因此受到越来越多人的关注。特别是一维最近邻的无规行走过程，由于它的理论表述上的简单性和数学上易处理性以及它对众多物理模型的代表性，而引起了人们的注意。本文主要研究的是存在有次近邻跳跃的随机行走问题，在次近邻跳跃的结果上取一些特殊值推导出最近邻跳跃的情况。
在这篇文章当中主要采用的研究方法是近年来量子行走中用到的方法。首先将主方程运用到简单一维复式格子中得到描写演化过程的矩阵，接着将描写演化的矩阵对角化再在K空间求解随机行走的演化问题，然后通过Fourier变换后回到坐标空间求解位移的平均和位移平方的平均值，最终得到在复式晶格上的随机行走中有次近邻跳跃的有意思的结果。
通过本文的计算和研究，当i无限大时得出了i步的随机行走，每一步的平均步长，其表达式为。它与i的奇偶性和L、S都无关，只与间距有关。并且知道在随机行走中的次近邻跳跃起到增加步长的作用。
关键词：一维复式晶格；次近邻跳跃；平均步长
Abstract:Random walk has important applications in many areas, and has been receiving more and more attention. Especially the random walk on one-dimensional lattice with nearest neighbor hopping, because of its mathematical simplicity and wide applications in many physical systems, has attracted much attention. In this paper we study the random walk on a one-dimensional non-Bravais lattice with next-nearest neighbor jumps.
In this article we utilize methods used in the study of quantum walk in recent years. First, the master equation is applied to one-dimensional non-Bravais lattice and the evolution is described by a matrix. Then the evolution matrix is diagonalized in the K space and then Fourier transformed back to coordinate space. The probability is used to solve the average displacement and the average squared displacement, and eventually we get the exact results on the random walk in the non-Bravais lattice with next-nearest neighbor jumps.
Through this calculation and study, we can see that when i is large, the behavior is the same with that of a simple random walk, each step of the average step size, with expression. It is independent of the parity of i nor the length of L 、S, only related to a and p. And the next-nearest neighbor jumps play the role of increasing the average step length.
Keywords: One-dimensional non-Bravais lattice; Next-nearest neighbor hopping; Average step length
