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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 andarXiv:2001.00303v3 fatcat:g4gzn2kobbgkhmbnff7qhgvawm