Universal record statistics for random walks and Lévy flights with a nonzero staying probability

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