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

arXiv: https://arxiv.org/abs/2001.00303 · HTML: https://mlsanity.com/paper/2001.00303/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jan 2020.

## Built on by (most cited first, 5 of 5)

- [Entropy contraction of the Gibbs sampler under log-concavity](https://mlsanity.com/paper/2410.00858/) (2024)
- [From Sampling to Optimization on Discrete Domains with Applications to Determinant Maximization](https://mlsanity.com/paper/2102.05347/) (2021)
- [Complexity of High-Dimensional Identity Testing with Coordinate Conditional Sampling](https://mlsanity.com/paper/2207.09102/) (2022)
- [Fast Doubly-Adaptive MCMC to Estimate the Gibbs Partition Function with Weak Mixing Time Bounds](https://mlsanity.com/paper/2111.07372/) (2021)
- [Rapid mixing in positively weighted restricted Boltzmann machines](https://mlsanity.com/paper/2604.00963/) (2026)

## Abstract (arXiv)

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 citeKM17,DK17,KO18,AL19, 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 citeWei06 for estimating the hardcore partition function, also answering a long-standing open problem of mixing time of Glauber dynamics citeLV97,LV99,DG00,Vig01,EHSVY16.