  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.
