Persistence of Random Walk Records
E. Ben-Naim and P.L. Krapivsky
We study records generated by Brownian particles in one
dimension. Specifically, we investigate an ordinary random walk and
define the record as the maximal position of the walk. We compare
the record of an individual random walk with the mean record,
obtained as an average over infinitely many realizations. We term
the walk ``superior'' if the record is always above average, and
conversely, the walk is said to be ``inferior'' if the record is
always below average. We find that the fraction of superior walks,
$S$, decays algebraically with time, $S\sim t^{-\beta}$, in the
limit $t\to\infty$, and that the persistence exponent is nontrivial,
$\beta=0.382258\ldots$. The fraction of inferior walks, $I$, also
decays as a power law, $I\sim t^{-\alpha}$, but the persistence
exponent is smaller, $\alpha=0.241608\ldots$. Both exponents are
roots of transcendental equations involving the parabolic cylinder
function. To obtain these theoretical results, we analyze the joint
density of superior walks with given record and position, while for
inferior walks it suffices to study the density as function of
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