The Asymptotics of Group Russian Roulette

We study the group Russian roulette problem, also known as the
shooting problem, defined as follows. We have n armed people in a room. At
each chime of a clock, everyone shoots a random other person. The persons shot
fall dead and the survivors shoot again at the next chime. Eventually, either
everyone is dead or there is a single survivor. We prove that the probability p_n
of having no survivors does not converge as n → ∞, and becomes asymptotically
periodic and continuous on the log n scale, with period 1.