Effective-medium approximation for lattice random walks with long-range jumps

Closed-form solutions to the dynamics of confined biased lattice random walks in arbitrary dimensions

Biased lattice random walks (BLRW) are used to model random motion with drift in a variety of empirical situations in engineering and natural systems such as phototaxis, chemotaxis, or gravitaxis. When motion is also affected by the presence of external borders resulting from natural barriers or experimental apparatuses, modelling biased random movement in confinement becomes necessary. To study these scenarios, confined BLRW models have been employed but so far only through computational techniques due to the lack of an analytic framework. Here, we lay the groundwork for such an analytical approach by deriving the Green's functions, or propagators, for the confined BLRW in arbitrary dimensions and arbitrary boundary conditions. By using these propagators we construct explicitly the time-dependent first-passage probability in one dimension for reflecting and periodic domains, while in higher dimensions we are able to find its generating function. The latter is used to find the mean first-passage passage time for a d-dimensional box, d-dimensional torus or a combination of both. We show the appearance of surprising characteristics such as the presence of saddles in the spatiotemporal dynamics of the propagator with reflecting boundaries, bimodal features in the first-passage probability in periodic domains and the minimization of the mean first-return time for a bias of intermediate strength in rectangular domains. Furthermore, we quantify how in a multitarget environment with the presence of a bias shorter mean first-passage times can be achieved by placing fewer targets close to boundaries in contrast to many targets away from them.

Reaction-diffusion on random spatial networks with scale-free jumping rates via effective medium theory

We study epidemic processes using a metapopulation approach on the line featuring random transport rates between arbitrarily distant sites. An average transport network is found using a recently developed variant of the effective medium approximation (EMA) that is capable of dealing with these long-range connections. Using a Feynman-Kac argument in the effective medium, we derive an estimate on the size of the infected domain, and reproduce the known result of its exponential growth in time. We hereby demonstrate the applicability of long-range EMA to dynamical processes on networks more intricate than simple diffusion.