Two Markov Models of the Spread of Rumors

We present two Markov models of the spreading of rumors. Specifically, we determine the limiting distribution as the population becomes large for the time to spreading of the rumor to the full population. For the first model, all particles are identical, but one initially knows the rumor. The limiting distribution is the convolution of two double exponential distributions. For the second model, there are two kinds of individuals: spreaders, who can spread the rumor, and ordinary individuals, who can only learn the rumor. Here, the limiting distribution is simply a double exponential distribution.