定向交换过程
The oriented swap process
摘要：
标记为$1,...,n$的颗粒最初以递增顺序排列。随后，当前处于递增顺序的每对相邻粒子根据速率为1的泊松过程交换。我们分析了这个过程的渐近行为为$n\to\infty$。我们证明了单个粒子的时空轨迹收敛（当适当缩放时）到具有两个不可微点的随机曲线族，并且置换矩阵在给定时刻收敛到具有绝对连续和奇异部分的确定性测度。在时间$(2+o(1))n$达到吸收状态（其中所有粒子按降序排列）。单个粒子的完成时间收敛到确定性的极限，波动渐近受特雷西-Widom分布。
英文摘要：
Particles labelled $1,...,n$ are initially arranged in increasing order. Subsequently, each pair of neighboring particles that is currently in increasing order swaps according to a Poisson process of rate 1. We analyze the asymptotic behavior of this process as $n\to\infty$. We prove that the space--time trajectories of individual particles converge (when suitably scaled) to a certain family of random curves with two points of non-differentiability, and that the permutation matrix at a given time converges to a certain deterministic measure with absolutely continuous and singular parts. The absorbing state (where all particles are in decreasing order) is reached at time $(2+o(1))n$. The finishing times of individual particles converge to deterministic limits, with fluctuations asymptotically governed by the Tracy--Widom distribution.
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