Virtually all Markov-chain Monte Carlo algorithms used for sampling a given distribution are reversible, and they satisfy the detailed-balance condition. For local chains, this leads to a slow, diffusive exploration of sample space. Significant speedups can be achieved through nonreversible algorithms with the given distribution as a targeted steady state. However, nonreversible algorithms for sampling are difficult to set up and to analyze, and exact speedup results for interacting many-particle systems are very rare. Here, we introduce the “lifted” totally asymmetric simple exclusion process (TASEP) as an exactly solvable paradigm for nonreversible many-particle Markov chains. It samples the same hard-sphere distribution as the Metropolis algorithm for symmetrically diffusing hard-core particles on a one-dimensional lattice. We solve the lifted TASEP by an unusual kind of coordinate Bethe ansatz and show that it exhibits polynomial (in particle number) speedups in the relaxation time for the asymptotic approach of the steady state, as well as the nonasymptotic mixing time, compared to both Metropolis and Kardar-Parisi-Zhang-based dynamics. The lifted TASEP is the reduction onto the one-dimensional lattice of the successful hard-sphere event-chain Monte Carlo algorithm, and we discuss that it can likewise be generalized to soft interaction potentials.

中文翻译：

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Lifted TASEP：加速多粒子马尔可夫链的可求解范式


几乎所有用于对给定分布进行采样的马尔可夫链蒙特卡洛算法都是可逆的，并且它们满足详细平衡条件。对于本地链，这会导致对样本空间的缓慢、扩散探索。通过不可逆算法，可以将给定的分布作为目标稳态，从而实现显著的加速。但是，用于采样的不可逆算法很难设置和分析，并且交互多粒子系统的精确加速结果非常罕见。在这里，我们介绍了“提升的”完全不对称简单排除过程 （TASEP） 作为不可逆多粒子马尔可夫链的精确可求解范式。它对与 Metropolis 算法相同的硬球分布进行采样，以便在一维晶格上对称扩散硬核粒子。我们通过一种不寻常的坐标 Bethe ansatz 求解了提升的 TASEP，并表明与基于 Metropolis 和 Kardar-Parisi-Zhang 的动力学相比，它在稳态渐近接近的弛豫时间以及非渐近混合时间中表现出多项式（粒子数）加速。提升的 TASEP 是成功的硬球事件链蒙特卡洛算法在一维晶格上的简化，我们讨论了它同样可以推广到软交互势。