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Exact record and order statistics of random walks via first-passage ideas

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  • Gregory Schehr
  • Satya N. Majumdar

Abstract

While records and order statistics of independent and identically distributed (i.i.d.) random variables X_1, ..., X_N are fully understood, much less is known for strongly correlated random variables, which is often the situation encountered in statistical physics. Recently, it was shown, in a series of works, that one-dimensional random walk (RW) is an interesting laboratory where the influence of strong correlations on records and order statistics can be studied in detail. We review here recent exact results which have been obtained for these questions about RW, using techniques borrowed from the study of first-passage problems. We also present a brief review of the well known (and not so well known) results for records and order statistics of i.i.d. variables.

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  • Gregory Schehr & Satya N. Majumdar, 2013. "Exact record and order statistics of random walks via first-passage ideas," Papers 1305.0639, arXiv.org.
    • Handle: RePEc:arx:papers:1305.0639
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    References listed on IDEAS

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    1. Krug, Joachim & Jain, Kavita, 2005. "Breaking records in the evolutionary race," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 358(1), pages 1-9.
    2. Le Doussal, Pierre & Monthus, Cécile, 2003. "Exact solutions for the statistics of extrema of some random 1D landscapes, application to the equilibrium and the dynamics of the toy model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 317(1), pages 140-198.
    3. Gregor Wergen & Satya N. Majumdar & Gregory Schehr, 2012. "Record Statistics for Multiple Random Walks," Papers 1204.5039, arXiv.org.
    4. P. Sibani, 2007. "Linear response in aging glassy systems, intermittency and the Poisson statistics of record fluctuations," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 58(4), pages 483-491, August.
    5. Gregor Wergen & Satya N. Majumdar & Gregory Schehr, 2012. "Record Statistics for Multiple Random Walks," Post-Print hal-00757701, HAL.
    6. Satya N. Majumdar & Philippe Mounaix & Gregory Schehr, 2013. "Exact Statistics of the Gap and Time Interval Between the First Two Maxima of Random Walks," Post-Print hal-00861416, HAL.
    7. Satya N. Majumdar & Gregory Schehr & Gregor Wergen, 2012. "Record statistics and persistence for a random walk with a drift," Post-Print hal-00744525, HAL.
    8. Satya N. Majumdar & Gregory Schehr & Gregor Wergen, 2012. "Record statistics and persistence for a random walk with a drift," Papers 1206.6972, arXiv.org, revised Aug 2012.
    9. Gregor Wergen & Miro Bogner & Joachim Krug, 2011. "Record statistics for biased random walks, with an application to financial data," Papers 1103.0893, arXiv.org.
    10. Satya N. Majumdar & Philippe Mounaix & Gregory Schehr, 2013. "Exact Statistics of the Gap and Time Interval Between the First Two Maxima of Random Walks," Papers 1303.4607, arXiv.org.
    11. Jean-Philippe Bouchaud & Marc Mezard, 1997. "Universality classes for extreme value statistics," Science & Finance (CFM) working paper archive 500043, Science & Finance, Capital Fund Management.
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    Cited by:

    1. Srivastava, Shashi C.L. & Lakshminarayan, Arul, 2015. "Records in the classical and quantum standard map," Chaos, Solitons & Fractals, Elsevier, vol. 74(C), pages 67-78.

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