Record Statistics for Multiple Random Walks

Gregor Wergen, Satya N. Majumdar and Gregory Schehr
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Gregor Wergen: Institut für Theoretische Physik [Köln] - Universität zu Köln = University of Cologne
Satya N. Majumdar: LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques - UP11 - Université Paris-Sud - Paris 11 - CNRS - Centre National de la Recherche Scientifique
Gregory Schehr: LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques - UP11 - Université Paris-Sud - Paris 11 - CNRS - Centre National de la Recherche Scientifique

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Abstract: We study the statistics of the number of records R_{n,N} for N identical and independent symmetric discrete-time random walks of n steps in one dimension, all starting at the origin at step 0. At each time step, each walker jumps by a random length drawn independently from a symmetric and continuous distribution. We consider two cases: (I) when the variance \sigma^2 of the jump distribution is finite and (II) when \sigma^2 is divergent as in the case of Lévy flights with index 0 1 in the two cases. We find that for large N, \alpha_N \approx 2 \sqrt{\log N} independently of \sigma^2 in case I. In contrast, in case II, the amplitude approaches to an N-independent constant for large N, \alpha_N \approx 4/\sqrt{\pi}, independently of 0

Date: 2012
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Citations: View citations in EconPapers (4)

Published in Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, 2012, 86, pp.011119. ⟨10.1103/PhysRevE.86.011119⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-00757701

DOI: 10.1103/PhysRevE.86.011119

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