Spectral Independence in High-Dimensional Expanders and Applications to the Hardcore Model

We say a discrete probability distribution over subsets of a finite ground set is spectrally independent if an associated pairwise influence matrix has a bounded largest eigenvalue for the distribution and all of its conditional distributions. We prove that if a distribution is spectrally independent, then the corresponding high dimensional simplicial complex is a local spectral expander. Using a line of recent works on mixing time of high dimensional walks on simplicial complexes [KM17]; [DK17]; [KO18]; [AL20], this implies that the corresponding Glauber dynamics mixes rapidly and generates (approximate) samples from the given distribution. As an application, we show that natural Glauber dynamics mixes rapidly (in polynomial time) to generate a random independent set from the hardcore model up to the uniqueness threshold. This improves the quasi-polynomial running time of Weitz's deterministic correlation decay algorithm [Wei06] for estimating the hardcore partition function, also answering a long-standing open problem of mixing time of Glauber dynamics [LV97]; [LV99]; [DG00]; [Vig01]; [Eft+16].