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Register a new userAsymptotic behavior of Aldous gossip process
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Aldous [(2007) Preprint] defined a gossip process in which space is a discrete $Ntimes N$ 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 $T=(2-2alpha/3)N^{alpha/3}log N$. 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 $T(varepsilon)approx T+N^{alpha/3}log (3varepsilon /M)$, 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.
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