  Aldous [(2007) Preprint] defined a gossip process in which space is a
discrete $N\times N$ torus, and the state of the process at time $t$ is the set
of individuals who know the information. Information spreads from a site to its
nearest neighbors at rate 1/4 each and at rate $N^{-\alpha}$ to a site chosen
at random from the torus. We will be interested in the case in which
$\alpha<3$, where the long range transmission significantly accelerates the
time at which everyone knows the information. We prove three results that
precisely describe the spread of information in a slightly simplified model on
the real torus. The time until everyone knows the information is asymptotically
$T=(2-2\alpha/3)N^{\alpha/3}\log N$. If $\rho_s$ is the fraction of the
population who know the information at time $s$ and $\varepsilon$ is small
then, for large $N$, the time until $\rho_s$ reaches $\varepsilon$ is
$T(\varepsilon)\approx T+N^{\alpha/3}\log (3\varepsilon /M)$, where $M$ is a
random variable determined by the early spread of the information. The value of
$\rho_s$ at time $s=T(1/3)+tN^{\alpha/3}$ is almost a deterministic function
$h(t)$ which satisfies an odd looking integro-differential equation. The last
result confirms a heuristic calculation of Aldous.
