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 P'{o}lya’s walk when the steps occur to either of the nearest-neighbouring sites. Since Smoluchowski’s 1906 derivation of the spatio-temporal 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 macro-molecules 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 modelling tool. Somewhat surprisingly, and differently from Brownian walks, the spatio-temporal 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 P'{o}lya’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 formulae to a single target in arbitrary dimensions are also presented. This in turn allows me 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 either 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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约束晶格随机游动在任意维度上的精确时空动力学：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时，通常称为P'{o} lya步行。自从Smoluchowski在1906年得出无界一维晶格中步行职业概率的时空相关性以来，离散的随机步行及其连续对应的Brownian步行在一个世纪的过程中发展成为广阔而广泛的知识领域。现在，通常会使用格子随机游走技术研究规模，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，与布朗人游走法不同，主要通过计算技术可以访问受限晶格游动概率的时空依赖性，发现其解析描述仍然是一个悬而未决的问题。利用一组具有Chebyshev多项式的解析组合恒等式，我开发了一种层次的维数约简，以发现受限P'{o} lya步态在任意维度上具有反射，周期性，沿每个方向吸收和混合（反射和吸收）边界条件。通过概率表达式，可以显式地一维地并通过在较高维度上进行积分来构造导出量的时间依赖性，例如对单个目标的首次通过概率，返回概率，所访问的不同站点的平均数量，不完全陷阱的吸收概率。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。反过来，这又使我能够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首次通过概率对许多目标中任一个的时间依赖性。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。