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：我们研究单点更新马尔可夫链（即Glauber动力学）在生成树上随机独立集时的混合时间。核心目标是为任意树结构获得最优收敛结果。我们考虑更一般的吉布斯分布采样问题，该问题在硬核模型中独立集由参数 $\lambda>0$ 加权。Martinelli、Sinclair 和 Weitz（2004）的先前工作对完全 $\Delta$ 正则树在所有 $\lambda$ 下获得了最优混合时间界。然而，Restrepo 等人（2014）表明，存在有界度树使得当 $\lambda$ 足够大时最优混合不成立。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。我们为任意树上未加权独立集的Glauber动力学建立了松驰时间（即逆谱隙）的最优界 $O(n)$。此外，当 $\lambda \leq 0.44$ 时，我们证明了最优混合时间界 $O(n \log n)$。需强调，我们的结果对任意树成立且不依赖于最大度 $\Delta$。有趣的是，我们的结果（远）超出了唯一性阈值（该阈值量级为 $\lambda = O(1/\Delta)$）。我们的证明方法受到近期谱独立性研究的启发。实际上，我们证明了任意树的谱独立性常数与最大度无关，但由于 Chen、Liu 和 Vigoda（2021）的最优混合结果仅适用于有界度图，因此这并不保证一般树上的混合性质。我们转而利用独立集的组合性质，通过非平凡的归纳证明直接导出方差/熵的近似张量化。