| Details: |
We study a rumour propagation model as a long-range percolation model on $\mathbb{Z}$. We begin by showing a sharp phase transition-type behaviour in terms of exponential decay of the rumour cluster in the sub-critical phase. In the super-critical phase, 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. We further show a central limit theorem for appropriately scaled and centered rightmost vertex.
We later introduce a rumour propagation model with re-activation and obtain the speed result
for the scaled rightmost vertex for this re-activated process as well. |