Tethering effects on first-passage variables of lattice random walks in linear and quadratic focal point potentials

Diffusion in a confining potential offers a minimal setting to understand the interplay between random motion and deterministic forces driving a particle towards a focal point or potential minimum. In continuous space and time, two extensively studied examples are Brownian motion in a linear () or a quadratic () potential. The deterministic bias towards the minimum is represented, respectively, by a constant force for the former and by an elastic restoring force that increases proportionally with distance for the latter. Surprisingly, unlike Brownian walks, random walks under focal point potentials in discrete space and time have received little attention. Here we bridge this gap by analyzing the dynamics of lattice random walkers in the presence of a potential, both in a finite and an infinite spatial domain, and a finite potential. For the V potential in unbounded space, we find the generating function of the occupation probability and analyze the time dependence of the mean number of distinct sites visited, demonstrating that its long-time growth is logarithmic. We also study the first-passage probability and show that its mean may display a minimum as a function of bias strength, depending on the location of the initial and target sites relative to the focal point. Qualitatively similar dependencies in the first-passage probability and its mean appear for the finite U potential. As a comparative analysis to the U potential, we construct the bounded V potential and superimpose in both cases a resetting process, in which the walker returns at random times to a site distinct from the focal point with some probability. We quantify the different effects of resetting on the steady-state probability and the first-passage dynamics in the two cases and show that a motion-limited regime emerges even for relatively moderate resetting probabilities.