# Record statistics for random walks and L\'evy flights with resetting

Abstract : We compute exactly the mean number of records $\langle R_N \rangle$ 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 $0\leq r < 1$) 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 $\langle R_N \rangle$ is completely universal, i.e., independent of the jump distribution $f(\eta)$. In particular, for large $N$, we show that $\langle R_N \rangle$ grows very slowly with increasing $N$ as $\langle R_N \rangle \approx (1/\sqrt{r})\, \ln N$ for \$0
Document type :
Journal articles
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https://hal.archives-ouvertes.fr/hal-03534126
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Submitted on : Wednesday, January 19, 2022 - 11:27:45 AM
Last modification on : Friday, January 21, 2022 - 4:13:25 AM

### Identifiers

• HAL Id : hal-03534126, version 1
• ARXIV : 2110.01539

### Citation

Satya N. Majumdar, Philippe Mounaix, Sanjib Sabhapandit, Gregory Schehr. Record statistics for random walks and L\'evy flights with resetting. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2022. ⟨hal-03534126⟩

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