Maxima of Two Random Walks: Universal Statistics of Lead Changes

We investigate statistics of lead changes of the maxima of two discrete-time random walks in one dimension. We show that the average number of lead changes grows as $\pi^{-1}\ln(t)$ 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 Levy flights. We also show that the probability to have at most n lead changes behaves as $t^{-1/4}[\ln t]^n$ for Brownian motion and as $t^{-\beta(\mu)}[\ln t]^n$ for symmetric Levy flights with index $\mu$. The decay exponent $\beta(\mu)$ varies continuously with the Levy index when $0<\mu<2$, while $\beta=1/4$ for $\mu>2$.