Gamblers ruin with many gamblers
Consider three gamblers with initial capital A,B,C. Each time, a pair of them is chosen at random, a fair coin is flipped and $1 is transferred. Classically, the chance that the first gambler wins all the money is A/ (A + B + C). Consider the first time one of the three goes broke. How long does this take, how is the money split between the other two (and how do these things depend on A,B,C)? These problems are seen as facts about rates of convergence to quasi-stationarity for Markov chains. In joint work with Kelsey Houston-Edwards and Laurent Saloff-Coste we have identified geometric conditions (John Domains, Whitney covers, inner regular domains) which allow fairly sharp analysis. This borrows heavily from geometry,PDE and harmonic analysis(Carleson estimates, parabolic Harnack inequalities). Adapting the tools to discrete problems is interesting work. I will try to explain all of the above 'in English'.