Record statistics for random walks and Levy flights with resetting

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 eta drawn independently from a symmetric and continuous distribution f (eta) 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 is an element of [0, 1 [ and any N, the mean number of records < R-N > is completely universal, i.e. independent of the jump distribution f (eta). In particular, for large N, we show that < R-N > grows very slowly with increasing N as < R-N > approximate to (1/root 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.