Access by Xinjiang University

Sparre-Andersen theorem with spatiotemporal correlations

Roberto Artuso^1,2,*, Giampaolo Cristadoro^3,†, Mirko Degli Esposti^3,‡, and Georgie Knight^3,§

¹Center for Nonlinear and Complex Systems, Università dell' Insubria, Via Valleggio 11, Como 22100, Italy
²I.N.F.N. Sezione di Milano, Via Celoria 16, Milano 20133, Italy
³Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy

^*roberto.artuso@uninsubria.it
^†giampaolo.cristadoro@unibo.it
^‡mirko.degliesposti@unibo.it
^§georgiesamuel.knight@unibo.it

Phys. Rev. E 89, 052111 – Published 8 May, 2014

DOI: https://doi.org/10.1103/PhysRevE.89.052111

Abstract

The Sparre-Andersen theorem is a remarkable result in one-dimensional random walk theory concerning the universality of the ubiquitous first-passage-time distribution. It states that the probability distribution $ρ_{n}$ of the number of steps needed for a walker starting at the origin to land on the positive semiaxes does not depend on the details of the distribution for the jumps of the walker, provided this distribution is symmetric and continuous, where in particular $ρ_{n} \sim n^{- 3 / 2}$ for large number of steps $n$ . On the other hand, there are many physical situations in which the time spent by the walker in doing one step depends on the length of the step and the interest concentrates on the time needed for a return, not on the number of steps. Here we modify the Sparre-Andersen proof to deal with such cases, in rather general situations in which the time variable correlates with the step variable. As an example we present a natural process in 2D that shows that deviations from normal scaling are present for the first-passage-time distribution on a semiplane.

Article Text

References (22)

S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).
S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, UK, 2001).
A. J. Bray, S. N. Majumdar, and G. Schehr, Adv. Phys. 62, 225 (2013).
E. Sparre Andersen, Math. Scand. 2, 195 (1954).
W. Feller, An Introduction to Probability Theory and Its Applications II (Wiley, New Jersey, USA, 1971).
J. Klafter and I. M. Sokolov, First Steps in Random Walks (Oxford University Press, Oxford, UK, 2011).
A. Baldassarri, J. P. Bouchaud, I. Dornic, and C. Godréche, Phys. Rev. E 59, R20 (1999); M. Bauer, C. Godréche, and J. M. Luck, J. Stat. Phys. 96, 963 (1999).
A. Zoia, A. Rosso, and S. N. Majumdar, Phys. Rev. Lett. 102, 120602 (2009).
H. Larralde, F. Leyvraz, G. Martinez-Mekler, R. Rechtman, and S. Ruffo, Phys. Rev. E 58, 4254 (1998).
D. Sornette, Critical Phenomena in Natural Sciences, 2nd ed. (Springer, Berlin, Germany, 2006).
P. Barthelemy, J. Bertolotti, and D. S. Wiersma, Nature (London) 453, 495 (2008).
E. W. Montroll and G. H. Weiss, J. Math. Phys. 6, 167 (1965); J. Klafter, A. Blumen, and M. F. Shlesinger, Phys. Rev. A 35, 3081 (1987); G. Zumofen and J. Klafter, Phys. Rev. E 47, 851 (1993).
R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).
I. M. Sokolov and R. Metzler, J. Phys. A: Math. Gen. 37, L609 (2004).
N. Korabel and E. Barkai, J. Stat. Mech.: Theory Exp. (2011) P05022.
B. Dybiec and E. Gudowska-Nowak, Europhys. Lett. 88, 10003 (2009).
E. Barkai, Phys. Rev. E 63, 046118 (2001).
R. Metzler and J. Klafter, Physica A 278, 107 (2000).
S. Denisov, V. Zaburdaev, and P. Hänggi, Phys. Rev. E 85, 031148 (2012).
S. N. Majumdar, Physica A 389, 4299 (2010); F. Pollaczek, Comptes Rendus 234, 2334 (1952); F. Spitzer, Trans. Amer. Math. Soc 82, 323 (1956); Duke Math. J. 24, 327 (1957).
Note that the choice of the distribution for the jumps is irrelevant as long as it is symmetric and continuous around $x = 0$ .
A. V. Chechkin, M. Hofmann, and I. M. Sokolov, Phys. Rev. E 80, 031112 (2009).

Physical Review E

Export citation

Export citation

Sparre-Andersen theorem with spatiotemporal correlations

Abstract

Article Text

References (22)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation