标题： 记录具有重置的随机游走和勒维飞行的统计信息 显示英文标题

标题： Record statistics for random walks and Lévy flights with resetting

摘要： 我们精确计算了大小为$N$的时间序列的记录平均数量$\langle R_N \rangle$，其中条目表示离散时间随机行走者在直线上的位置。 在每个时间步，行走者以长度$\eta$跳跃，该长度独立地从对称且连续的分布$f(\eta)$中抽取，概率为$1-r$（其中$0\leq r < 1$），而以互补概率$r$，它会重置到起始点$x=0$。 这是一个弱相关时间序列的精确可解示例，它在强相关随机游走序列（对于$r=0$）和不相关时间序列（对于$(1-r) \ll 1$）之间进行插值。值得注意的是，我们发现对于每个固定的$r \in [0,1[$和任何$N$，记录的平均数量$\langle R_N \rangle$完全具有普遍性，即与跳跃分布$f(\eta)$无关。 特别是，对于较大的$N$，我们证明当$N$增加时，$\langle R_N \rangle$增长得非常缓慢，正如$\langle R_N \rangle \approx (1/\sqrt{r})\, \ln N$对于$0<r <1$。 我们还计算了$\langle R_N \rangle$在两个极限$r \to 0$和$r \to 1$的精确通用交叉标度函数。 我们的分析预测与数值模拟结果高度一致。 显示英文摘要

摘要： 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.