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 $\lambda>0$; the special case $\lambda=1$ corresponds to the uniform distribution over all independent sets. 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. 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 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）的混合时间，该过程用于生成树的随机独立集。我们的核心目标是为任意树获得最优收敛结果。我们考虑更一般的问题：在硬核模型中从吉布斯分布采样，其中独立集由参数$\lambda>0$加权；特殊情况$\lambda=1$对应所有独立集的均匀分布。Martinelli、Sinclair和Weitz（2004）的前期工作对完全$\Delta$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，当$\lambda$足够大时，存在有界度树使得最优混合不成立。Eppstein和Frishberg（2022）近期工作证明了任意树上Glauber动力学的多项式混合时间界，并推广至有界树宽图。我们建立了任意树上无权重独立集的Glauber动力学弛豫时间（即逆谱间隙）的最优界$O(n)$。需强调，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（大幅）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。实际上，我们证明了任意树的谱独立性成立且常数与最大度无关，但这并不直接蕴含一般树的混合性，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性质，通过非平凡的归纳证明直接建立方差的近似张量化。