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

Nima Anari, Kuikui Liu, Shayan Oveis Gharan
2020 arXiv   pre-print
We say a probability distribution μ is spectrally independent if an associated correlation matrix has a bounded largest eigenvalue for the distribution and all of its conditional distributions. We prove that if μ 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 , this implies that the corresponding Glauber dynamics mixes rapidly and
more » ... rates (approximate) samples from μ. 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 for estimating the hardcore partition function, also answering a long-standing open problem of mixing time of Glauber dynamics .
arXiv:2001.00303v3 fatcat:g4gzn2kobbgkhmbnff7qhgvawm