  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.
