Maxima of two random walks: Universal statistics of lead changes
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Boston Univ., Boston, MA (United States); Univ. Paris-Saclay and CNRS, Gif-sur-Yvette (France)
- Univ. Paris-1 Pantheon-Sorbonne, Paris (France)
In this study, 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 $${\pi }^{-1}\mathrm{ln}t$$ 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 $${t}^{-1/4}{(\mathrm{ln}t)}^{n}$$ for Brownian motion and as $${t}^{-\beta (\mu )}{(\mathrm{ln}t)}^{n}$$ for symmetric Lévy flights with index μ. The decay exponent $$\beta \equiv \beta (\mu )$$ varies continuously with the Lévy index when $$0\lt \mu \lt 2$$, and remains constant $$\beta =1/4$$ for $$\mu \gt 2$$.
- Research Organization:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Sponsoring Organization:
- USDOE
- Grant/Contract Number:
- AC52-06NA25396
- OSTI ID:
- 1255157
- Report Number(s):
- LA-UR-15-29655
- Journal Information:
- Journal of Physics. A, Mathematical and Theoretical, Vol. 49, Issue 20; ISSN 1751-8113
- Publisher:
- IOP PublishingCopyright Statement
- Country of Publication:
- United States
- Language:
- English
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