Optimal mixing of the down-up walk on independent sets of a given size
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Let G be a graph on n vertices of maximum degree Δ. We show that, for any δ > 0, the down-up walk on independent sets of size k ≤ (1-δ)α_c(Δ)n mixes in time O_Δ,δ(klogn), thereby resolving a conjecture of Davies and Perkins in an optimal form. Here, α_c(Δ)n is the NP-hardness threshold for the problem of counting independent sets of a given size in a graph on n vertices of maximum degree Δ. Our mixing time has optimal dependence on k,n for the entire range of k; previously, even polynomial mixing was not known. In fact, for k = Ω_Δ(n) in this range, we establish a log-Sobolev inequality with optimal constant Ω_Δ,δ(1/n). At the heart of our proof are three new ingredients, which may be of independent interest. The first is a method for lifting ℓ_∞-independence from a suitable distribution on the discrete cube – in this case, the hard-core model – to the slice by proving stability of an Edgeworth expansion using a multivariate zero-free region for the base distribution. The second is a generalization of the Lee-Yau induction to prove log-Sobolev inequalities for distributions on the slice with considerably less symmetry than the uniform distribution. The third is a sharp decomposition-type result which provides a lossless comparison between the Dirichlet form of the original Markov chain and that of the so-called projected chain in the presence of a contractive coupling.
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