The small world effect on the coalescing time of random walks

TL;DR

This paper analyzes how adding random shortcuts to a d-dimensional torus affects the meeting time of two random walks, showing they meet faster on small worlds for dimensions 1 and 2, with dimension-dependent effects for higher dimensions.

Contribution

It provides the asymptotic behavior of the coalescing time on small worlds, extending known results from regular tori and identifying the impact of random connections.

Findings

Random walks meet faster on small worlds for d ≤ 2.

The rescaling factor for meeting time is proportional to L^d on small worlds.

For d ≥ 3, the meeting time depends on the probability of using random connections.

Abstract

A small world is obtained from the $d$-dimensional torus of size 2L adding randomly chosen connections between sites, in a way such that each site has exactly one random neighbour in addition to its deterministic neighbours. We study the asymptotic behaviour of the meeting time $T_L$ of two random walks moving on this small world and compare it with the result on the torus. On the torus, in order to have convergence, we have to rescale $T_L$ by a factor $C_1L^2$ if $d=1$, by $C_2L^2\log L$ if $d=2$ and $C_dL^d$ if $d\ge3$. We prove that on the small world the rescaling factor is $C^\prime_dL^d$ and identify the constant $C^\prime_d$, proving that the walks always meet faster on the small world than on the torus if $d\le2$, while if $d\ge3$ this depends on the probability of moving along the random connection. As an application, we obtain results on the hitting time to the origin of a single walk and on the convergence of coalescing random walk systems on the small world.