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 $k \leq (1-\delta)\alpha_c(\Delta)n$ 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.

翻译：设$G$是最大度为$\Delta$的$n$顶点图。我们证明，对于任意$\delta > 0$，大小为$k \leq (1-\delta)\alpha_c(\Delta)n$的独立集上的下-上行走在时间$O_{\Delta,\delta}(k\log{n})$内混合，从而以最优形式解决了Davies和Perkins的猜想。这里，$\alpha_{c}(\Delta)n$是在最大度为$\Delta$的$n$顶点图中计算给定大小独立集数量的NP难度阈值。我们的混合时间对$k$的整个范围具有关于$k,n$的最优依赖性；此前，甚至多项式混合也未知。事实上，对于该范围内$k = \Omega_{\Delta}(n)$的情形，我们建立了具有最优常数$\Omega_{\Delta,\delta}(1/n)$的对数Sobolev不等式。我们证明的核心包含三个可能具有独立意义的新要素。第一个是一种方法，用于将$\ell_\infty$-独立性从离散立方体上的适当分布——在此情形下为硬核模型——提升到切片上，通过使用基础分布的多元零区域证明Edgeworth展开的稳定性。第二个是Lee-Yau归纳法的推广，用于证明切片上比均匀分布对称性显著更低的分布的对数Sobolev不等式。第三个是锐利分解型结果，在存在收缩耦合的情况下，提供原始马尔可夫链的Dirichlet形式与所谓投影链的Dirichlet形式之间的无损比较。