Maxima of two random walks: Universal statistics of lead changes
In this study, we investigate statistics of lead changes of the maxima of two discretetime random walks in one dimension. We show that the average number of lead changes grows as $${\pi }^{1}\mathrm{ln}t$$ in the longtime 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 $$\mu \gt 2$$.
 Authors:

^{[1]};
^{[2]};
^{[3]}
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Boston Univ., Boston, MA (United States); Univ. ParisSaclay and CNRS, GifsurYvette (France)
 Univ. Paris1 PantheonSorbonne, Paris (France)
 Publication Date:
 Report Number(s):
 LAUR1529655
Journal ID: ISSN 17518113
 Grant/Contract Number:
 AC5206NA25396
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of Physics. A, Mathematical and Theoretical
 Additional Journal Information:
 Journal Volume: 49; Journal Issue: 20; Journal ID: ISSN 17518113
 Publisher:
 IOP Publishing
 Research Org:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org:
 USDOE
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING
 OSTI Identifier:
 1255157
BenNaim, E., Krapivsky, P. L., and RandonFurling, J.. Maxima of two random walks: Universal statistics of lead changes. United States: N. p.,
Web. doi:10.1088/17518113/49/20/205003.
BenNaim, E., Krapivsky, P. L., & RandonFurling, J.. Maxima of two random walks: Universal statistics of lead changes. United States. doi:10.1088/17518113/49/20/205003.
BenNaim, E., Krapivsky, P. L., and RandonFurling, J.. 2016.
"Maxima of two random walks: Universal statistics of lead changes". United States.
doi:10.1088/17518113/49/20/205003. https://www.osti.gov/servlets/purl/1255157.
@article{osti_1255157,
title = {Maxima of two random walks: Universal statistics of lead changes},
author = {BenNaim, E. and Krapivsky, P. L. and RandonFurling, J.},
abstractNote = {In this study, we investigate statistics of lead changes of the maxima of two discretetime random walks in one dimension. We show that the average number of lead changes grows as ${\pi }^{1}\mathrm{ln}t$ in the longtime 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 $\mu \gt 2$.},
doi = {10.1088/17518113/49/20/205003},
journal = {Journal of Physics. A, Mathematical and Theoretical},
number = 20,
volume = 49,
place = {United States},
year = {2016},
month = {4}
}