Record statistics for random walks and Lévy flights with resetting

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. 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 $0\le r<1$) 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 $(1-r)\ll 1$). Remarkably, we found that for every fixed $r\in [0,1[$ 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 $⟨R_N⟩\approx (1/\sqrt{r})\ln N$ for $0<r<1$. We also computed the exact universal crossover scaling functions for $⟨R_N⟩$ in the two limits $r\to 0$ and $r\to 1$. Our analytical predictions are in excellent agreement with numerical simulations.

24 pages, 7 figures. Version submitted for publication

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