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 $\delta\gt0$, the down-up walk on independent sets of size $k \leq(1-\delta) \alpha_{c}(\Delta) n$ mixes in time $O_{\Delta, \delta}(k \log n)$, 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 $\ell_{\infty}$-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.