Record statistics for random walks and Lévy flights with resetting

Abstract

Abstract We compute exactly the mean number of records ⟨R N ⟩ for a time-series of size N whose entries represent the positions of a discrete time random walker on the line with resetting. At each time step, the walker jumps by a length η drawn independently from a symmetric and continuous distribution f(η) with probability 1 − r (with 0 ⩽ r < 1) and with the complementary probability r it resets to its starting point x = 0. This is an exactly solvable example of a weakly correlated time-series that interpolates between a strongly correlated random walk series (for r = 0) and an uncorrelated time-series (for (1 − r) ≪ 1). Remarkably, we found that for every fixed r ∈ 0 , 1 and any N, the mean number of records ⟨R N ⟩ is completely universal, i.e. independent of the jump distribution f(η). In particular, for large N, we show that ⟨R N ⟩ grows very slowly with increasing N as ⟨ R N ⟩ ≈ ( 1 / r ) ln N for 0 < r < 1. We also computed the exact universal crossover scaling functions for ⟨R N ⟩ in the two limits r → 0 and r → 1. Our analytical predictions are in excellent agreement with numerical simulations.