Maxima of Two Random Walks: Universal Statistics of Lead Changes

Abstract

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 $\mu >2$.

7 pages, 6 figures

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