A Tale of Two Walks: Kipnis, Marchioro and Presutti Meet Kac in a Quantum World
Abstract
We reveal an unexpected connection between two interacting particle systems: Kac's walk and the Kipnis-Marchioro-Presutti (KMP) process. We first establish a mixing-time analogue of Aldous's conjecture for the KMP process on general graphs: the mixing time of the t-particle process on any connected d-vertex graph is at most O(log d) times the corresponding single-particle mixing time, uniformly in t. As an application, we improve the mixing-time bound for coordinate hit-and-run on the n-dimensional standard simplex from Õ(n3) [KV26] to O(nlog n), while removing the dependence on the initial distribution. We then show that the twirling channel induced by the parallel Kac's walk on the symmetric subspace can be exactly encoded by a Markov chain on partitions, which naturally lifts to a parallel KMP chain on the complete graph. We prove that O(log d+log(1/ε)) repetitions of the parallel Kac's walk suffice to approximate Haar twirling on the symmetric subspace of (ℂd)⊗ t to error ε, uniformly in all t. Our main technical contribution is a new structural property of KMP-type dynamics: conditional product structure, which holds for both the standard KMP processes on general graphs and the parallel KMP processes on complete graphs. Conditioned on suitable auxiliary randomness, the evolution of the labeled KMP process factorizes into independent single-particle updates. Combining this structure with an exact coupling inspired by the analysis of Kac's walk, we remove the dependence on the number of particles from the mixing bounds for both unlabeled KMP models. This bypasses the log(πmin−1) dependency from spectral-gap arguments, which is often prohibitively large for many-particle systems. Using this framework, we prove mixing bounds that are sharp up to logarithmic factors for both unlabeled KMP models.
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BibTeX
@misc{chen2026tale,
title = {{A Tale of Two Walks: Kipnis, Marchioro and Presutti Meet Kac in a Quantum World}},
author = {Qian Chen and Jingcheng Liu and Minglong Qin and Leonard Schulman and Fang Song and Penghui Yao and Mingnan Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38044},
eprint = {2609.38044},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38044},
}
RIS
TY - UNPB
TI - A Tale of Two Walks: Kipnis, Marchioro and Presutti Meet Kac in a Quantum World
AU - Qian Chen
AU - Jingcheng Liu
AU - Minglong Qin
AU - Leonard Schulman
AU - Fang Song
AU - Penghui Yao
AU - Mingnan Zhao
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38044
UR - https://arxiv.org/abs/2609.38044
AB - We reveal an unexpected connection between two interacting particle systems: Kac's walk and the Kipnis-Marchioro-Presutti (KMP) process. We first establish a mixing-time analogue of Aldous's conjecture for the KMP process on general graphs: the mixing time of the t-particle process on any connected d-vertex graph is at most O(log d) times the corresponding single-particle mixing time, uniformly in t. As an application, we improve the mixing-time bound for coordinate hit-and-run on the n-dimensional standard simplex from Õ(n^3) [KV26] to O(nlog n), while removing the dependence on the initial distribution. We then show that the twirling channel induced by the parallel Kac's walk on the symmetric subspace can be exactly encoded by a Markov chain on partitions, which naturally lifts to a parallel KMP chain on the complete graph. We prove that O(log d+log(1/ε)) repetitions of the parallel Kac's walk suffice to approximate Haar twirling on the symmetric subspace of (ℂ^d)^⊗ t to error ε, uniformly in all t. Our main technical contribution is a new structural property of KMP-type dynamics: conditional product structure, which holds for both the standard KMP processes on general graphs and the parallel KMP processes on complete graphs. Conditioned on suitable auxiliary randomness, the evolution of the labeled KMP process factorizes into independent single-particle updates. Combining this structure with an exact coupling inspired by the analysis of Kac's walk, we remove the dependence on the number of particles from the mixing bounds for both unlabeled KMP models. This bypasses the log(π_min^−1) dependency from spectral-gap arguments, which is often prohibitively large for many-particle systems. Using this framework, we prove mixing bounds that are sharp up to logarithmic factors for both unlabeled KMP models.
ER -