Satya N. Majumdar 1 Philippe Mounaix 2 Sanjib Sabhapandit 3 Gregory Schehr 4
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⟩
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
- 1. LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques
- 2. CPHT – Centre de Physique Théorique [Palaiseau]
- 3. Raman Research Institute
- 4. LPTHE – Laboratoire de Physique Théorique et Hautes Energies