标题： 布朗运动重排的顶级统计量的普遍性 显示英文标题

标题： Universality of Top Rank Statistics for Brownian Reshuffling

摘要： 我们研究了在半线上进行布朗运动的粒子的最高排名统计量的动力学方面，这些粒子按其距离原点的距离进行排序。 为此，我们引入了一个称为重叠比$\Omega(t)$的可观测量，其平均值是某个时刻在 top-$n$列表中的粒子在时间$t$后仍位于 top-$n$列表中的概率。 重叠比是一个局部可观测量，它集中在排名的顶部，并不需要对所有粒子进行完整排序。 在实践中很容易测量。 我们推导出一个系统在稳态下具有独立布朗运动的$N$个粒子的平均重叠比的解析公式，这些粒子在原点处有一个反射墙，并朝向墙的方向移动。 特别是，我们证明对于$N\rightarrow \infty$，重叠比率呈现出一种相当简单的形式$\langle \Omega(t)\rangle = {\rm erfc}(a \sqrt{t})$，对于$n\gg 1$，其中包含某个缩放参数$a>0$。 这个结果即使对于中等大小的前$n$列表，例如$n=10$，也是一个很好的近似。 此外，如我们所展示的，重叠比率表现出在许多动力系统中观察到的普遍行为，包括几何布朗运动、具有位置依赖漂移和一侧有软屏障的布朗运动、Bouchaud-Mézard财富分布模型以及Kesten过程。 显示英文摘要

摘要： We study the dynamical aspects of the top rank statistics of particles, performing Brownian motions on a half-line, which are ranked by their distance from the origin. For this purpose, we introduce an observable that we call the overlap ratio $\Omega(t)$, whose average is the probability that a particle that is on the top-$n$ list at some time will also be on the top-$n$ list after time $t$. The overlap ratio is a local observable which is concentrated at the top of the ranking and does not require the full ranking of all particles. It is simple to measure in practice. We derive an analytical formula for the average overlap ratio for a system of $N$ particles in the stationary state that undergo independent Brownian motion on the positive real half-axis with a reflecting wall at the origin and a drift towards the wall. In particular, we show that for $N\rightarrow \infty$, the overlap ratio takes a rather simple form $\langle \Omega(t)\rangle = {\rm erfc}(a \sqrt{t})$ for $n\gg 1$ with some scaling parameter $a>0$. This result is a very good approximation even for moderate sizes of the top-$n$ list such as $n=10$. Moreover, as we show, the overlap ratio exhibits universal behavior observed in many dynamical systems including geometric Brownian motion, Brownian motion with a position-dependent drift and a soft barrier on one side, the Bouchaud-M\'ezard wealth distribution model, and Kesten processes.