Discrete space-time resetting model: application to first-passage and transmission statistics

We consider the dynamics of lattice random walks with resetting. The walker moving randomly on a lattice of arbitrary dimensions resets at every time step to a given site with a constant probability r. We construct a discrete renewal equation and present closed-form expressions for different quantities of the resetting dynamics in terms of the underlying reset-free propagator or Green's function. We apply our formalism to the biased random walk dynamics in one-dimensional (1D) unbounded space and show how one recovers in the continuous limits results for diffusion with resetting. The resetting dynamics of biased random walker in 1D domain bounded with periodic and reflecting boundaries is also analyzed. Depending on the bias the first-passage probability in periodic domain shows multi-fold non-monotonicity as r is varied. Finally, we apply our formalism to study the transmission dynamics of two lattice walkers with resetting in 1D domain bounded by periodic and reflecting boundaries. The probability of a definite transmission between the walkers shows non-monotonic behavior as the resetting probabilities are varied.

Automated summary

This paper investigates lattice random walks with resetting dynamics. By constructing a discrete renewal equation and deriving closed-form expressions, the authors provide a formalism for analyzing various quantities in resetting dynamics based on the reset-free propagator or Green's function. The formalism is applied to biased random walks in one-dimensional unbounded space, and the continuous limits yield results consistent with diffusion with resetting. The paper also explores the resetting dynamics of biased random walkers with periodic and reflecting boundaries, and observes non-monotonic behavior in the first-passage probability in periodic domains as the resetting probability varies. Additionally, the authors study the transmission dynamics of two lattice walkers with resetting in a one-dimensional domain bounded by periodic and reflecting boundaries, and find non-monotonic behavior in the probability of definite transmission as the resetting probabilities change.