In the field of sampling algorithms, MCMC (Markov Chain Monte Carlo) methods are widely used when direct sampling is not possible. However, multimodality of target distributions often leads to slow convergence and mixing. One common solution is parallel tempering. Though highly effective in practice, theoretical guarantees on its performance are limited. In this paper, we present a new lower bound for parallel tempering on the spectral gap that has a polynomial dependence on all parameters except $\log L$, where $(L + 1)$ is the number of levels. This improves the best existing bound which depends exponentially on the number of modes. Moreover, we complement our result with a hypothetical upper bound on spectral gap that has an exponential dependence on $\log L$, which shows that, in some sense, our bound is tight.

翻译：在采样算法领域，当直接取样不可行时，MCMC（马尔可夫链蒙特卡罗）方法被广泛使用。然而目标分布的多模性经常导致收敛速度慢和混合效果不佳问题。一种常见的解决方案是并行温度。虽然在实践中高度有效，但对其性能的理论保证有限。在本文中，我们提出了一种新的并行温度的下限谱隙界限，其关于所有参数都有多项式依赖，仅仅除去以 $(L+1)$ 表示的级数数目的 $\log L$。这改进了现有最佳界限，后者在模数的数量上呈指数关系。此外，我们补充了一个假设上限，其关于 $\log L$ 具有指数依赖性，这表明我们的下限是紧的。(注：英文中的Proper Nouns与中文中的专有名词并不完全一致，此处忽略)