Record statistics for random walks and levy flights with resetting

Abstract

We compute exactly the mean number of records ⟨RN⟩ 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 $r\in \left[\right.0,1\left[\right.$ and any N, the mean number of records ⟨RN⟩ is completely universal, i.e. independent of the jump distribution f(η). In particular, for large N, we show that ⟨RN⟩ grows very slowly with increasing N as $\langle {R}_{N}\rangle \approx (1/\sqrt{r})\mathrm{ln}\enspace N$ for 0 < r < 1. We also computed the exact universal crossover scaling functions for ⟨RN⟩ in the two limits r → 0 and r → 1. Our analytical predictions are in excellent agreement with numerical simulations.