Top rank statistics for Brownian reshuffling

We study the dynamical aspects of the top rank statistics of particles, performing Brownian motions on a half-line, which are ranked by their distance from the origin. For this purpose, we introduce an observable Ω(t) which we call the overlap ratio. The average overlap ratio is equal to the probability that a particle that is on the top-n list at some time will also be on the top-n list after time t. The overlap ratio is a local observable which is concentrated at the top of the ranking and does not require the full ranking of all particles. In practice, the overlap ratio is easy to measure. We derive an analytical formula for the average overlap ratio for a system of N particles in the stationary state that undergo independent Brownian motion on the positive real half-axis with a reflecting wall at the origin and a drift towards the wall. In particular, we show that for N→∞, the overlap ratio takes a rather simple form 〈Ω(t)〉=erfc(asqrt[t]) for n≫1 with some scaling parameter a>0. This result is a very good approximation even for moderate sizes of the top-n list such as n=10. Moreover, we observe in numerical studies that the overlap ratio exhibits universal behavior in many dynamical systems including geometric Brownian motion, Brownian motion with asymptotically linear drift, the Bouchaud-Mézard wealth distribution model, and Kesten processes. We conjecture the universality to hold for a broad class of one-dimensional stochastic processes.