Multi-target Search in Bounded and Heterogeneous Environments: A Lattice Random Walk Perspective

For more than a century lattice random walks have been employed ubiquitously, both as a theoretical laboratory to develop intuition about more complex stochastic processes and as a tool to interpret a vast array of empirical observations. Recent advances in lattice random walk theory in bounded and heterogeneous environments have opened up opportunities to cope with the finely resolved spatio-temporal nature of modern movement data. We review such advances and their formalisms to represent analytically the walker spatio-temporal dynamics in arbitrary dimensions and geometries. As new findings, we derive the exact spatio-temporal representation of biased walks in a periodic hexagon, we use the discrete Feynman–Kac equation to describe a walker’s interaction with a radiation boundary and we unearth a disorder indifference phenomenon. To demonstrate the power of the formalism for first-passage processes we uncover the appearance of multiple first-passage peaks with biased walkers in a periodic hexagon, we display the dependence of the first-transmission probability on the proximity transfer efficiency between two resetting walkers in a one-dimensional periodic lattice, we present an example of spatial disorder in a two-dimensional square lattice that strongly affects the splitting probabilities to either of two targets and we study the first-reaction dynamics to a single lattice site in an unbounded one-dimensional lattice.