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

TL;DR

This paper investigates lattice random walks under focal point potentials, analyzing first-passage properties, effects of resetting, and differences between linear and quadratic potentials in finite and infinite domains.

Contribution

It provides the first detailed analysis of discrete lattice random walks in focal point potentials, including generating functions, first-passage probabilities, and resetting effects.

Findings

Long-time growth of visited sites is logarithmic in unbounded V-potential.

Mean first-passage time can have a minimum depending on bias strength.

Resetting induces a motion-limited regime even at moderate probabilities.

Abstract

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 (V-shaped) or a quadratic (U-shaped) 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 analysing the dynamics of lattice random walkers in the presence of a V-shaped potential, both in a finite and an infinite spatial domain, and a finite U-shaped potential. For the V-potential in unbounded space, we find the generating function of the occupation probability and analyse 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 a motion-limited regime emerges even for relatively moderate resetting probabilities.