This paper initiates a limit theory of permutation valued processes, building on the recent theory of permutons. We apply this to study the asymptotic behaviour of random sorting networks. We prove that the Archimedean path, the conjectured limit of random sorting networks, is the unique path from the identity to the reverse permuton having minimal energy in an appropriate metric. Together with a recent large deviations result (Kotowski, 2016), it implies the Archimedean limit for the model of relaxed random sorting networks.

中文翻译：

---

排列极限的几何

本文以最近的置换理论为基础，提出了置换价值过程的极限理论。我们将此应用于研究随机排序网络的渐近行为。我们证明了阿基米德路径，即随机排序网络的推测极限，是从身份到反向置换的唯一路径，在适当的度量中具有最小能量。连同最近的大偏差结果（Kotowski，2016），它暗示了松弛随机排序网络模型的阿基米德极限。