Synchronization of random walks with reflecting boundaries

TL;DR

This paper analytically investigates the synchronization process of two reflecting boundary random walks, deriving formulas for mean synchronization time, probability distribution, and ensemble behavior, with implications for neural synchronization.

Contribution

It provides the first analytical calculation of synchronization time and probability distribution for reflecting boundary random walks, linking these results to neural synchronization phenomena.

Findings

Synchronization time scales with the square of system size

Synchronization probability converges to a geometric distribution

Ensemble synchronization time grows logarithmically with size

Abstract

Reflecting boundary conditions cause two one-dimensional random walks to synchronize if a common direction is chosen in each step. The mean synchronization time and its standard deviation are calculated analytically. Both quantities are found to increase proportional to the square of the system size. Additionally, the probability of synchronization in a given step is analyzed, which converges to a geometric distribution for long synchronization times. From this asymptotic behavior the number of steps required to synchronize an ensemble of independent random walk pairs is deduced. Here the synchronization time increases with the logarithm of the ensemble size. The results of this model are compared to those observed in neural synchronization.