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A large deviations principle for the Maki-Thompson rumour model

We consider the stochastic model for the propagation of a rumour within a population which was formulated by Maki and Thompson. Sudbury established that, as the population size tends to infinity, the proportion of the population never hearing the rumour converges in probability to $0.2032$. Watson later derived the asymptotic normality of a suitably scaled version of this proportion. We prove a corresponding large deviations principle, with an explicit formula for the rate function.