We study the mixing time of the single-site update Markov chain, known as the Glauber dynamics, for generating a random independent set of a tree. Our focus is obtaining optimal convergence results for arbitrary trees. We consider the more general problem of sampling from the Gibbs distribution in the hard-core model where independent sets are weighted by a parameter $\lambda>0$. Previous work of Martinelli, Sinclair and Weitz (2004) obtained optimal mixing time bounds for the complete $\Delta$-regular tree for all $\lambda$. However, Restrepo et al. (2014) showed that for sufficiently large $\lambda$ there are bounded-degree trees where optimal mixing does not hold. Recent work of Eppstein and Frishberg (2022) proved a polynomial mixing time bound for the Glauber dynamics for arbitrary trees, and more generally for graphs of bounded tree-width. We establish an optimal bound on the relaxation time (i.e., inverse spectral gap) of $O(n)$ for the Glauber dynamics for unweighted independent sets on arbitrary trees. Moreover, for $\lambda\leq .44$ we prove an optimal mixing time bound of $O(n\log{n})$. We stress that our results hold for arbitrary trees and there is no dependence on the maximum degree $\Delta$. Interestingly, our results extend (far) beyond the uniqueness threshold which is on the order $\lambda=O(1/\Delta)$. Our proof approach is inspired by recent work on spectral independence. In fact, we prove that spectral independence holds with a constant independent of the maximum degree for any tree, but this does not imply mixing for general trees as the optimal mixing results of Chen, Liu, and Vigoda (2021) only apply for bounded degree graphs. We instead utilize the combinatorial nature of independent sets to directly prove approximate tensorization of variance/entropy via a non-trivial inductive proof.

翻译：我们研究单点位更新马尔可夫链（即Glauber动力学）在生成树上的随机独立集时的混合时间。我们的重点是为任意树获取最优收敛结果。我们考虑更一般的问题：在硬核模型中，独立集由参数 $\lambda>0$ 加权，从Gibbs分布中采样。Martinelli、Sinclair和Weitz（2004）的前期工作为所有 $\lambda$ 下的完全 $\Delta$-正则树获得了最优混合时间界。然而，Restrepo等人（2014）表明，对于足够大的 $\lambda$，存在有界度树使得最优混合不成立。Eppstein和Frishberg（2022）的最新工作证明了任意树（以及更一般的树宽有界图）上Glauber动力学的多项式混合时间界。我们为任意树上无加权独立集的Glauber动力学建立了松弛时间（即逆谱隙）的最优界 $O(n)$。此外，对于 $\lambda\leq 0.44$，我们证明了最优混合时间界 $O(n\log{n})$。我们强调，我们的结果适用于任意树，且不依赖于最大度 $\Delta$。有趣的是，我们的结果（远远）超越了唯一性阈值（约为 $\lambda=O(1/\Delta)$）。我们的证明方法受近期关于谱独立性的工作启发。事实上，我们证明谱独立性在任意树上以独立于最大度的常数成立，但这并不意味着对一般树有混合性质，因为Chen、Liu和Vigoda（2021）的最优混合结果仅适用于有界度图。我们转而利用独立集的组合特性，通过非平凡的归纳证明，直接建立方差/熵的近似张量化。