Statistics of the Number of Records for Random Walks and Lévy Flights on a $1D$ Lattice

Abstract

We study the statistics of the number of records $R_n$ for a symmetric, $n$-step, discrete jump process on a $1D$ lattice. At a given step, the walker can jump by arbitrary lattice units drawn from a given symmetric probability distribution. This process includes, as a special case, the standard nearest neighbor lattice random walk. We derive explicitly the generating function of the distribution $P(R_n)$ of the number of records, valid for arbitrary discrete jump distributions. As a byproduct, we provide a relatively simple proof of the generalized Sparre Andersen theorem for the survival probability of a random walk on a line, with discrete or continuous jump distributions. For the discrete jump process, we then derive the asymptotic large $n$ behavior of $P(R_n)$ as well as of the average number of records $E(R_n)$. We show that unlike the case of random walks with symmetric and continuous jump distributions where the record statistics is strongly universal (i.e., independent of the jump distribution for all $n$), the record statistics for lattice walks depends on the jump distribution for any fixed $n$. However, in the large $n$ limit, we show that the distribution of the scaled record number $R_n/E(R_n)$ approaches a universal, half-Gaussian form for any discrete jump process. The dependence on the jump distribution enters only through the scale factor $E(R_n)$, which we also compute in the large $n$ limit for arbitrary jump distributions. We present explicit results for a few examples and provide numerical checks of our analytical predictions.

34 pages, 6 figures

References in corpus (12)

• Universal Record Statistics of Random Walks and Lévy Flights
• Universal survival probability for a $d$-dimensional run-and-tumble particle
• Record statistics for biased random walks, with an application to financial data
• Record statistics and persistence for a random walk with a drift
• Record dynamics and the observed temperature plateau in the magnetic creep rate of type II superconductors
• Records in a changing world
• Scaling in Tournaments
• Record Statistics for Multiple Random Walks
• Record occurrence and record values in daily and monthly temperatures
• A record-driven growth process
• Record Statistics of Continuous Time Random Walk
• One- and two-sample nonparametric tests for the signal-to-noise ratio based on record statistics

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• A continuous-time random walk extension of the Gillis model
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