Record-breaking statistics for random walks in the presence of measurement error and noise

Yaniv Edery 1, Alexander B. Kostinski 2, Satya N. Majumdar 3, Brian Berkowitz 1

Physical Review Letters 110 (2013) 180602

We address the question of distance record-setting by a random walker in the presence of measurement error, $\delta$, and additive noise, $\gamma$ and show that the mean number of (upper) records up to $n$ steps still grows universally as $< R_n> \sim n^{1/2}$ for large $n$ for all jump distributions, including Lévy flights, and for all $\delta$ and $\gamma$. In contrast to the universal growth exponent of 1/2, the pace of record setting, measured by the pre-factor of $n^{1/2}$, depends on $\delta$ and $\gamma$. In the absence of noise ($\gamma=0$), the pre-factor $S(\delta)$ is evaluated explicitly for arbitrary jump distributions and it decreases monotonically with increasing $\delta$ whereas, in case of perfect measurement $(\delta=0)$, the corresponding pre-factor $T(\gamma)$ increases with $\gamma$. Our analytical results are supported by extensive numerical simulations and qualitatively similar results are found in two and three dimensions.

  • 1 : Department of Environmental Sciences and Energy Research
    Weizmann Institute of Science,
  • 2 : Department of Physics
    Michigan Technological University
  • 3 : Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS)
    CNRS : UMR8626 – Université Paris XI - Paris Sud