  We say a probability distribution $\mu$ 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 $\mu$
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
\cite{KM17,DK17,KO18,AL19}, this implies that the corresponding Glauber
dynamics mixes rapidly and generates (approximate) samples from $\mu$.
  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 \cite{Wei06} for
estimating the hardcore partition function, also answering a long-standing open
problem of mixing time of Glauber dynamics \cite{LV97,LV99,DG00,Vig01,EHSVY16}.
