First encounters on Bethe Lattices and 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.

25 pages, 11 figures

References in corpus (4)

• Survival probability of a particle in a sea of mobile traps: A tale of tails
• Trapping in dendrimers and regular hyperbranched polymers
• Influence of trap location on the efficiency of trapping in dendrimers and regular hyperbranched polymers
• Mean trapping time for an arbitrary node on regular hyperbranched polymers

Cited by in corpus (3)

• Analytical results for the distribution of first return times of random walks on random regular graphs
• The joint distribution of first return times and of the number of distinct sites visited by a 1D random walk before returning to the origin
• Analytical results for the distribution of first return times of non-backtracking random walks on configuration model networks