Statistics of the number of records for random walks and Lévy flights on a 1D lattice
Philippe Mounaix, Satya N. Majumdar, Grégory Schehr
Abstract
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.