两种行走的故事：Kipnis、Marchioro 与 Presutti 在量子世界中邂逅 Kac

中文摘要 AI 辅助

我们揭示了并行Kac行走与Kipnis-Marchioro-Presutti（KMP）过程之间意想不到的联系。由并行Kac行走在对称子空间上诱导的twirling信道恰好被一个在分拆上的经典马尔可夫链所编码，该链提升为完全图上的并行KMP过程。这一对应关系将twirling信道的分析简化为并行KMP过程的混合。我们证明，$O(\log d+\log(1/\varepsilon))$次重复足以在$(\mathbb C^d)^{\otimes t}$的对称子空间上以误差$\varepsilon$近似Haar twirling，且该界对副本数$t$一致成立。对于一般图上的标准KMP过程，我们证明了Aldous猜想的一个混合时间类比：在固定精度下，$t$粒子过程的混合时间至多为单粒子混合时间乘以顶点数对数的常数倍，且对$t$一致成立。作为应用，我们将$n$维标准单纯形上坐标hit-and-run的全变差混合时间界从$\widetilde O(n^3)$（Kook和Vempala，2026）改进为$\widetilde O(n)$，同时去除对初始分布的依赖。我们的主要技术贡献是并行和标准KMP过程的条件乘积结构。在适当的辅助随机性条件下，标记粒子独立演化。将该结构与精确耦合相结合，为两种无标记KMP模型导出了对粒子数一致的混合界。这些界在对数因子内是紧的，并意味着并行Kac twirling信道在对称子空间上的快速收敛。

英文摘要

We reveal an unexpected connection between the parallel Kac's walk and the Kipnis-Marchioro-Presutti (KMP) process. The twirling channel induced by the parallel Kac's walk on the symmetric subspace is exactly encoded by a classical Markov chain on partitions, which lifts to a parallel KMP process on complete graphs. This correspondence reduces the analysis of the twirling channel to the mixing of the parallel KMP process. We prove that $O(\log d+\log(1/\varepsilon))$ repetitions suffice to approximate Haar twirling on the symmetric subspace of $(\mathbb C^d)^{\otimes t}$ to error $\varepsilon$, uniformly in the number of copies $t$. For the standard KMP process on general graphs, we prove a mixing-time analogue of Aldous's conjecture: at fixed accuracy, the mixing time of the $t$-particle process is at most a constant times the single-particle mixing time multiplied by the logarithm of the number of vertices, uniformly in $t$. As an application, we improve the total variation mixing-time bound for coordinate hit-and-run on the $n$-dimensional standard simplex from $\widetilde O(n^3)$ (Kook and Vempala, 2026) to $\widetilde O(n)$, while removing the dependence on the initial distribution. Our main technical contribution is conditional product structure for both parallel and standard KMP processes. Conditioned on suitable auxiliary randomness, the labeled particles evolve independently. Combining this structure with an exact coupling yields mixing bounds uniform in the number of particles for both unlabeled KMP models. These bounds are sharp up to logarithmic factors and imply rapid convergence of the parallel Kac twirling channel on the symmetric subspace.

发表机构

• The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
• Nanjing University（南京大学）
• National University of Singapore（新加坡国立大学）
• California Institute of Technology（加州理工学院）
• Portland State University（波特兰州立大学）
• Hefei National Laboratory（合肥国家实验室）

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