Abstract
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´olya's walk when the steps occur to either of the nearestneighbouring sites. Since Smoluchowski's 1906 derivation of the spatiotemporal dependence of the walk occupation probability in an unbounded onedimensional 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 onedimensional search of proteins along a DNA strand and the twodimensional roaming of bacteria in a petri dish, to the threedimensional 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 modelling 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 P´olya'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 timedependence of derived quantities, explicitly in onedimension and via an integration in higher dimensions, such as the firstpassage probability to a single target, return probability, average number of distinct sites visited, and absorption probability with imperfect traps. Exact mean firstpassage time formulae to a single target in arbitrary dimensions are also presented. This in turn allows me to extend the socalled discrete pseudoGreen function formalism, employed to determine analytically mean firstpassage 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 firstpassage probability to either of many targets. Reduction of the occupation probability expressions to the continuous time limit, the socalled continuous time random walk, and to the spacetime continuous limit is also presented.
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 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 P´olya'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 timedependence of derived quantities, explicitly in onedimension and via an integration in higher dimensions, such as the firstpassage probability to a single target, return probability, average number of distinct sites visited, and absorption probability with imperfect traps. Exact mean firstpassage time formulae to a single target in arbitrary dimensions are also presented. This in turn allows me to extend the socalled discrete pseudoGreen function formalism, employed to determine analytically mean firstpassage 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 firstpassage probability to either of many targets. Reduction of the occupation probability expressions to the continuous time limit, the socalled continuous time random walk, and to the spacetime continuous limit is also presented.
Original language  English 

Article number  021045 
Number of pages  20 
Journal  Physical Review X 
Volume  10 
Issue number  2 
DOIs  
Publication status  Published  28 May 2020 
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Dr Luca Giuggioli
 Department of Engineering Mathematics  Associate Professor in Complexity Sciences
 Applied Nonlinear Mathematics
 Animal Behaviour and Sensory Biology
 Ecology and Environmental Change
Person: Academic , Member