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

TL;DR

This paper derives an exact, universal formula for the average number of records in a resetting random walk, revealing a slow logarithmic growth with the number of steps and showing universality across jump distributions.

Contribution

It provides the first exact solution for record statistics in a resetting random walk, demonstrating universality and deriving crossover scaling functions.

Findings

Mean record number grows as (1/√r) log N for large N.

Record statistics are universal, independent of jump distribution.

Exact crossover functions for different resetting probabilities.

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 $0<r <1$. We also computed the exact universal crossover scaling functions for $\langle R_N \rangle$ in the two limits $r \to 0$ and $r \to 1$. Our analytical predictions are in excellent agreement with numerical simulations.