First encounters on Bethe Lattices and Cayley Trees

TL;DR

This paper analyzes the first encounter probabilities between a target and a trap on Bethe Lattices and Cayley trees, revealing how target movement and initial positions influence survival times and introducing a new simulation method.

Contribution

It provides analytical and numerical insights into encounter dynamics on Bethe Lattices and Cayley trees, and introduces a memory-efficient simulation method for random walks.

Findings

Fixed targets prolong survival on Bethe Lattices.

Target movement can extend survival on Cayley trees depending on initial position.

New memory-based simulation method for random walks on Cayley trees.

Abstract

In this work we consider the first encounter problems between a fixed and/or mobile target A and a moving trap B on Bethe Lattices and Cayley trees. The survival probability (SP) of the target A on the both kinds of structures are analyzed analytically and compared. On Bethe Lattices, the results show that the fixed target will still prolong its survival time, whereas, on Cayley trees, there are some initial positions where the target should move to prolong its survival time. The mean first encounter time (MFET) for mobile target A is evaluated numerically and compared with the mean first passage time (MFPT) for the fixed target A. Different initial settings are addressed and clear boundaries are obtained. These findings are helpful for optimizing the strategy to prolong the survival time of the target or to speed up the search process on Cayley trees, in relation to the target's movement and the initial position configuration of the two walkers. We also present a new method, which uses a small amount of memory, for simulating random walks on Cayley trees.