# Comparison of two models of tethered motion

## TL;DR

This paper compares two models of tethered random walk behavior, analyzing their dynamics and first-passage properties, revealing how different return mechanisms affect search efficiency and survival probabilities in complex environments.

## Contribution

It introduces a unified analytical framework for comparing biased and resetting tethered walkers, deriving key quantities like first-passage times and splitting probabilities.

## Key findings

- Different return mechanisms significantly affect search times.
- Non-monotonic effects observed in survival probabilities.
- Analytical results applicable to biological and physical systems.

## Abstract

We consider a random walker whose motion is tethered around a focal point. We use two models that exhibit the same spatial dependence in the steady state but widely different dynamics. In one case, the walker is subject to a deterministic bias towards the focal point, while in the other case, it resets its position to the focal point at random times. The deterministic tendency of the biased walker makes the forays away from the focal point more unlikely when compared to the random nature of the returns of the resetting walker. This difference has consequences on the spatio-temporal dynamics at intermediate times. To show the differences in the two models, we analyze their probability distribution and their dynamics in presence and absence of partially or fully absorbing traps. We derive analytically various quantities: (i) mean first-passage times to one target, where we recover results obtained earlier by a different technique, (ii) splitting probabilities to either of two targets as well as survival probabilities when one or either target is partially absorbing. The interplay between confinement, diffusion and absorbing traps produces interesting non-monotonic effects in various quantities, all potentially accessible in experiments. The formalism developed here may have a diverse range of applications, from study of animals roaming within home ranges and of electronic excitations moving in organic crystals to developing efficient search algorithms for locating targets in a crowded environment.