Universität zu Köln

Majumdar, Satya N., Schehr, Gregory and Wergen, Gregor (2012). Record statistics and persistence for a random walk with a drift. J. Phys. A-Math. Theor., 45 (35). BRISTOL: IOP PUBLISHING LTD. ISSN 1751-8121

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Abstract

We study the statistics of records of a one-dimensional random walk of n steps, starting from the origin, and in the presence of a constant bias c. At each time step, the walker makes a random jump of length. drawn from a continuous distribution f(eta), which is symmetric around a constant drift c. We focus in particular on the case where f (eta) is a symmetric stable law with a Levy index 0 < mu <= 2. The record statistics crucially depends on the persistence probability, which, as we show here, exhibits different behaviors depending on the sign of c and the value of the parameter mu. Hence, in the limit of a large number of steps n, the record statistics is sensitive to these parameters (c and mu) of the jump distribution. We compute the asymptotic mean record number < R-n > after n steps as well as its full distribution P (R, n). We also compute the statistics of the ages of the longest and the shortest lasting record. Our exact computations show the existence of five distinct regions in the (c, 0 < mu <= 2) strip where these quantities display qualitatively different behaviors. We also present numerical simulation results that verify our analytical predictions.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Majumdar, Satya N. UNSPECIFIED UNSPECIFIED UNSPECIFIED Schehr, Gregory UNSPECIFIED UNSPECIFIED UNSPECIFIED Wergen, Gregor UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-483166
DOI:	10.1088/1751-8113/45/35/355002
Journal or Publication Title:	J. Phys. A-Math. Theor.
Volume:	45
Number:	35
Date:	2012
Publisher:	IOP PUBLISHING LTD
Place of Publication:	BRISTOL
ISSN:	1751-8121
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language ANOMALOUS DIFFUSION; EVOLUTION Multiple languages Physics, Multidisciplinary; Physics, Mathematical Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/48316