Maxima of Two Random Walks : Universal Statistics of Lead Changes
We investigate statistics of lead changes of the maxima of two random walks in one dimension. We show that the average number of lead changes grows as (1/π) ln(t) in the long-time limit. We present theoretical and numerical evidence that this asymptotic behavior is universal. Specifically, this behavior is independent of the jump distribution : the same asymptotic underlies standard Brownian motion and symmetric Lévy flights. We also show that the probability to have at most n lead changes behaves as t^(−1/4) [lnt]^n for Brownian motion and as t^(−β(μ)) [lnt]^n for symmetric Lévy flights with index μ. The decay exponent β(μ) varies continuously with the Lévy index when 0<μ<2.
[Work done with E. Ben-Naim (Los Alamos National Laboratory) and Paul Krapivsky (Boston University)]