Universal record statistics for random walks and L\'evy flights with a nonzero staying probability

TL;DR

This paper provides an exact, universal analysis of record statistics in a discrete-time random walk with a nonzero staying probability, revealing anti-correlations and a universal scaling limit as the staying probability approaches one.

Contribution

It introduces a universal framework for record statistics in random walks with staying probability, independent of jump distribution, and explores the effects of staying probability on correlations and fluctuations.

Findings

Record statistics are universal and independent of jump distribution for all staying probabilities.

Nonzero staying probability induces anti-correlations between record events.

The Fano factor decreases with increasing staying probability, indicating reduced fluctuations.

Abstract

We compute exactly the statistics of the number of records in a discrete-time random walk model on a line where the walker stays at a given position with a nonzero probability $0\leq p \leq 1$, while with the complementary probability $1-p$, it jumps to a new position with a jump length drawn from a continuous and symmetric distribution $f_0(\eta)$. We have shown that, for arbitrary $p$, the statistics of records up to step $N$ is completely universal, i.e., independent of $f_0(\eta)$ for any $N$. We also compute the connected two-time correlation function $C_p(m_1, m_2)$ of the record-breaking events at times $m_1$ and $m_2$ and show it is also universal for all $p$. Moreover, we demonstrate that $C_p(m_1, m_2)< C_0(m_1, m_2)$ for all $p>0$, indicating that a nonzero $p$ induces additional anti-correlations between record events. We further show that these anti-correlations lead to a drastic reduction in the fluctuations of the record numbers with increasing $p$. This is manifest in the Fano factor, i.e. the ratio of the variance and the mean of the record number, which we compute explicitly. We also show that an interesting scaling limit emerges when $p \to 1$, $N \to \infty$ with the product $t = (1-p)\, N$ fixed. We compute exactly the associated universal scaling functions for the mean, variance and the Fano factor of the number of records in this scaling limit. .