Maxima of Two Random Walks: Universal Statistics of Lead Changes
E. Ben-Naim and P.L. Krapivsky
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 L\'evy 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 L\'evy flights with index $\mu$. The decay
exponent $\beta\equiv \beta(\mu)$ varies continuously with the L\'evy
index when $0<\mu<2$, while $\beta=1/4$ for $\mu>2$.
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