UVictoria Probability and Dynamics Seminar: Eric Foxall

We consider the model in which uniform random points are added to the unit interval at a constant intensity and independently vanish each at rate 1. The stationary distribution is a Poisson point process. Our goal is to investigate the time until an atypically large gap appears, in the high-intensity limit. To do so we develop some theory that allows us to compute the hitting time of a rare set in a family of Markov chains in terms of the restriction of the stationary distribution to that set. By studying the stationary distribution, as well as sample paths of the counting processes that describe particle numbers on fixed intervals, we obtain an asymptotic formula for the expected time until the appearance of a large gap. A component of the theory involves generalizing the exponential limit of scaled geometric random variables to the case where the relevant Bernoulli sequence is one-dependent.