Exact Spatiotemporal Dynamics of Confined Lattice Random Walks in Arbitrary Dimensions: A Century after Smoluchowski and Pólya

摘要

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 nearestneighbor 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 firstpassage 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.