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

Abstract

We say a probability distribution 𝜇 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 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 [T. Kaufman and D. Mass, Proceedings of ITCS, 2017, pp. 4:1–4:27; I. Dinur and T. Kaufman, Proceedings of the IEEE 58th Annual Symposium on Foundations of Computer Science, 2017, pp. 974–985; T. Kaufman and I. Oppenheim, Proceedings of APPROX/RANDOM, 2018, pp. 47:1–47:17; V. L. Alev and L. C. Lau, Proceedings of the 52nd Annual ACM Symposium on Theory of Computing, 2020], this implies that the corresponding Glauber dynamics mixes rapidly and generates (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 [D. Weitz, Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 140–149] for estimating the hardcore partition function, also answering a long-standing open problem of mixing time of Glauber dynamics [M. Luby and E. Vigoda, Proceedings of the 29th Annual ACM Symposium on Theory of Computing, 1997, pp. 682–687; M. Luby and E. Vigoda, Random Structures Algorithms, 15 (1999), pp. 229–241; M. Dyer and C. Greenhill, J. Algorithms, 35 (2000), pp. 17–49; E. Vigoda, Electron. J. Combin., 8 (2001); C. Efthymiou et al., Proceedings of FOCS, 2016, pp. 704–713].