论文标题

随机步行的最大和最小步行之间的时间分布

Distribution of the Time Between Maximum and Minimum of Random Walks

论文作者

Mori, Francesco, Majumdar, Satya N., Schehr, Gregory

论文摘要

我们认为固定持续时间$ t $的一维布朗尼运动。使用路径积分技术,我们准确地计算了差异$τ= t _ {\ min} -t _ {\ max} $之间的时间$ t _ {\ min} $之间的全球最小值$ t _ {\ min} $。我们将此结果扩展到布朗桥,即周期性的布朗尼时期$ t $。在这两种情况下,我们都会通过分析$τ$的第一瞬间计算,以及$ t _ {\ max} $和$ t _ {\ min} $的协方差,表明这些时间是反相关的。我们证明,Brownian Motion的$τ$的分布对于具有$ n $ steps的离散时间随机步行和有限的跳跃差异有效,以限制$ n \ to \ infty $。对于具有不同跳跃差异的Lévy航班,我们数值验证了$τ$的分布是否与布朗案例不同。对于带有连续和对称跳跃的随机步行,我们实际上验证了事件的概率“ $τ= n $”的概率正好是任何有限$ n $的$ 1/(2n)$,独立于跳跃分布。我们的结果还可以应用于描述$(1+1)$ - 尺寸固定状态的Kardar-Parisi-Zhang界面之间的最大高度和最小高度之间的距离,而在有限尺寸$ l $的基​​板上生长。我们的发现通过数值模拟证实。这些结果中的一些已在最近的一封信中宣布[Phys。莱特牧师。 123,200201(2019)]。

We consider a one-dimensional Brownian motion of fixed duration $T$. Using a path-integral technique, we compute exactly the probability distribution of the difference $τ=t_{\min}-t_{\max}$ between the time $t_{\min}$ of the global minimum and the time $t_{\max}$ of the global maximum. We extend this result to a Brownian bridge, i.e. a periodic Brownian motion of period $T$. In both cases, we compute analytically the first few moments of $τ$, as well as the covariance of $t_{\max}$ and $t_{\min}$, showing that these times are anti-correlated. We demonstrate that the distribution of $τ$ for Brownian motion is valid for discrete-time random walks with $n$ steps and with a finite jump variance, in the limit $n\to \infty$. In the case of Lévy flights, which have a divergent jump variance, we numerically verify that the distribution of $τ$ differs from the Brownian case. For random walks with continuous and symmetric jumps we numerically verify that the probability of the event "$τ= n$" is exactly $1/(2n)$ for any finite $n$, independently of the jump distribution. Our results can be also applied to describe the distance between the maximal and minimal height of $(1+1)$-dimensional stationary-state Kardar-Parisi-Zhang interfaces growing over a substrate of finite size $L$. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 123, 200201 (2019)].

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