Title: On rumour propagation among sceptics,
Abstract: Junior, Machado and Zuluaga (2011) studied a model to understand the spread
of a rumour. Their model consists of individuals situated at the integer points
of the line $\N$. An individual at the origin $0$ starts a rumour and passes it
to all individuals in the interval $[0,R_0]$, where $R_0$ is a non-negative
random variable. An individual located at $i$ in this interval receives the
rumour and transmits it further among individuals in $[i, i+R_i]$ where $R_0$
and $R_i$ are i.i.d. random variables. The rumour spreads in this manner. An
alternate model considers individuals seeking to find the rumour from
individuals who have already heard it. For this s/he asks individuals to the
left of her/him and lying in an interval of a random size. We study these two
models, when the individuals are more sceptical and they transmit or accept the
rumour only if they receive it from at least two different sources.
In stochastic geometry the equivalent of this rumour process is the study of
coverage of the space $\N^d$ by random sets. Our study here extends the study
of coverage of space and considers the case when each vertex of $\N^d$ is
covered by at least two distinct random sets.