Optimal Mixing via Tensorization for Random Independent Sets on Arbitrary Trees
CoRR(2023)
摘要
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 λ>0. Previous
work of Martinelli, Sinclair and Weitz (2004) obtained optimal mixing time
bounds for the complete Δ-regular tree for all λ. However,
Restrepo et al. (2014) showed that for sufficiently large λ 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 λ≤ .44 we prove an optimal mixing
time bound of O(nlogn). We stress that our results hold for arbitrary
trees and there is no dependence on the maximum degree Δ. Interestingly,
our results extend (far) beyond the uniqueness threshold which is on the order
λ=O(1/Δ). 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.
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关键词
optimal mixing,random independent sets,tensorization,trees
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