How fast do rumours spread?

We study a rumour propagation model along the lines of \cite{lebensztayn2008disk} as a long-range percolation model on \(\Z\). We begin by showing a sharp phase transition-type behaviour in the sense of exponential decay of the survival time of the rumour cluster in the sub-critical phase. In the super-critical phase, \update{under the assumption that radius of influence r.v. has \(2+\epsilon\) moment finite (for some \(\epsilon>0\))}, we show that the rightmost vertex in the rumour cluster has a deterministic speed in the sense that after appropriate scaling, the location of the rightmost vertex converges a.s.\ to a deterministic positive constant. \update{Under the assumption that radius of influence r.v. has \(4+\epsilon\) moment finite,} we obtain a central limit theorem for appropriately scaled and centred rightmost vertex. Later, we introduce a rumour propagation model with reactivation. For this section, we work with a family of exponentially decaying i.i.d. radius of influence r.v.'s, and we obtain the speed result for the scaled rightmost position of the rumour cluster. Each of these results is novel, in the sense that such properties have never been established before in the context of the rumour propagation model on \(\Z\), to the best of our knowledge.