Optimal mixing of the down-up walk on independent sets of a given size

Let G be a graph on n vertices of maximum degree Delta. We show that, for any delta > 0, the down-up walk on independent sets of size k <= (1 - delta)alpha(c)(Delta)n mixes in time O-Delta,O-delta(k log n), thereby resolving a conjecture of Davies and Perkins in an optimal form. Here, alpha(c)(Delta)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 Delta. 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 = Omega(Delta)(n) in this range, we establish a log-Sobolev inequality with optimal constant Omega(Delta,delta)(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(infinity)-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.