A lattice random walk is a mathematical representation of movement through random steps on a lattice at discrete times. It is commonly referred to as Pólya’s walk when the steps occur in either of the nearest-neighbor sites. Since Smoluchowski’s 1906 derivation of the spatiotemporal dependence of the walk occupation probability in an unbounded one-dimensional lattice, discrete random walks and their continuous counterpart, Brownian walks, have developed over the course of a century into a vast and versatile area of knowledge. Lattice random walks are now routinely employed to study stochastic processes across scales, dimensions, and disciplines, from the one-dimensional search of proteins along a DNA strand and the two-dimensional roaming of bacteria in a petri dish, to the three-dimensional motion of macromolecules inside cells and the spatial coverage of multiple robots in a disaster area. In these realistic scenarios, when the randomly moving object is constrained to remain within a finite domain, confined lattice random walks represent a powerful modeling tool. Somewhat surprisingly, and differently from Brownian walks, the spatiotemporal dependence of the confined lattice walk probability has been accessible mainly via computational techniques, and finding its analytic description has remained an open problem. Making use of a set of analytic combinatorics identities with Chebyshev polynomials, I develop a hierarchical dimensionality reduction to find the exact space and time dependence of the occupation probability for confined Pólya’s walks in arbitrary dimensions with reflective, periodic, absorbing, and mixed (reflective and absorbing) boundary conditions along each direction. The probability expressions allow one to construct the time dependence of derived quantities, explicitly in one dimension and via an integration in higher dimensions, such as the first-passage probability to a single target, return probability, average number of distinct sites visited, and absorption probability with imperfect traps. Exact mean first-passage time formulas to a single target in arbitrary dimensions are also presented. These formulas allow one to extend the so-called discrete pseudo-Green function formalism, employed to determine analytically mean first-passage time, with reflecting and periodic boundaries, and a wealth of other related quantities, to arbitrary dimensions. For multiple targets, I introduce a procedure to construct the time dependence of the first-passage probability to one of many targets. Reduction of the occupation probability expressions to the continuous time limit, the so-called continuous time random walk, and to the space-time continuous limit is also presented.

中文翻译：

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任意尺度上有限格子随机游动的精确时空动力学：Smoluchowski和Pólya之后的一个世纪

晶格随机游动是在离散时间通过晶格上的随机步长运动的数学表示。当台阶发生在任何最近的站点中时，通常称为Pólya步行。自从Smoluchowski在1906年推导了无界一维晶格中步行职业概率的时空依赖性之后，一个世纪以来，离散的随机步行及其连续对应的Brownian步行已发展成为一个广阔而广泛的知识领域。现在，通常会采用随机抽样的方式来研究跨尺度，规模和学科的随机过程，从沿着DNA链的一维蛋白质搜索到培养皿中细菌的二维漫游，单元内部大分子的三维运动以及灾区多个机器人的空间覆盖。在这些现实情况中，当将随机移动的对象约束为保留在有限域内时，受限晶格随机游动表示一种强大的建模工具。出乎意料的是，与布朗步行不同，局限的格子步行概率的时空相关性主要是通过计算技术获得的，发现其解析描述仍然是一个悬而未决的问题。利用Chebyshev多项式的一组解析组合恒等式，我开发了一个层次的维数约简，以发现受限的Pólya步道在任意维度上具有反射，周期性，吸收，并沿每个方向混合（反射和吸收）边界条件。通过概率表达式，可以明确地在一个维度上并通过在更高维度上的积分来构造导出量的时间依赖性，例如对单个目标的初次通过概率，返回概率，所访问的不同位点的平均数目以及吸收陷阱不完善的可能性。还给出了任意维度上单个目标的精确平均首次通过时间公式。这些公式使人们可以将所谓的离散伪格林函数形式主义扩展到任意维度，该形式用于确定具有反射和周期性边界以及大量其他相关量的分析平均首次通过时间。对于多个目标，我介绍了一种程序来构造第一次通过概率对许多目标之一的时间依赖性。还提出了将占用概率表达式减少到连续时间极限，所谓的连续时间随机游走以及时空连续极限的方法。