The small world effect on the coalescing time of random walks

A small world is obtained from the dd-dimensional torus of size 2L2L 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 TLTL 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 TLTL by a factor C1L2C1L2 if d=1d=1, by C2L2logLC2L2logL if d=2d=2 and CdLdCdLd if d≥3d≥3. We prove that on the small world the rescaling factor is Cd′Ld and identify the constant Cd′, proving that the walks always meet faster on the small world than on the torus if d≤2d≤2, while if d≥3d≥3 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.