连续时间Mallows过程的全局与局部极限
The global and local limit of the continuous-time Mallows process
AI总结：
本文研究连续时间Mallows过程的渐近性质，证明当n趋于无穷时该过程存在基于permuton极限的全局极限和与整数集Mallows分布相关的局部极限，揭示其与随机排序网络极限性质的相似性。
摘要：
连续时间Mallows过程是集合$\{1, \ldots, n\}$上的随机排列过程，其在时刻$t$的边缘分布为参数$t$的Mallows分布。近期Corsini证明存在唯一的马尔可夫Mallows过程，其左逆序是独立计数过程。我们证明当$n \to \infty$时，该过程存在全局极限与局部极限。全局极限通过对空间和时间进行适当尺度变换得到，是定义在$[0,1]$上的显式随机过程，其描述基于Starr分析的Mallows分布的permuton（排列子）极限。局部极限是一个$\mathbb{Z}$上的排列过程，与Gnedin和Olshanski提出的$\mathbb{Z}$上排列的Mallows分布构造密切相关。我们的结果表明，Mallows过程的渐近行为与近期研究的随机排序网络的极限性质具有相似性。
英文摘要：
Continuous-time Mallows processes are processes of random permutations of the set $\{1, \ldots, n\}$ whose marginal at time $t$ is the Mallows distribution with parameter $t$. Recently Corsini showed that there exists a unique Markov Mallows process whose left inversions are independent counting processes. We prove that this process admits a global and a local limit as $n \to \infty$. The global limit, obtained after suitably rescaling space and time, is an explicit stochastic process on $[0,1]$ whose description is based on the permuton limit of the Mallows distribution, analyzed by Starr. The local limit is a process of permutations of $\mathbb{Z}$ which is closely related to the construction of the Mallows distribution on permutations of $\mathbb{Z}$ due to Gnedin and Olshanski. Our results demonstrate an analogy between the asymptotic behavior of Mallows processes and the recently studied limiting properties of random sorting networks.
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