  We introduce a notion called entropic independence that is an entropic analog
of spectral notions of high-dimensional expansion. Informally, entropic
independence of a background distribution $\mu$ on $k$-sized subsets of a
ground set of elements says that for any (possibly randomly chosen) set $S$,
the relative entropy of a single element of $S$ drawn uniformly at random
carries at most $O(1/k)$ fraction of the relative entropy of $S$. Entropic
independence is the analog of the notion of spectral independence, if one
replaces variance by entropy. We use entropic independence to derive tight
mixing time bounds, overcoming the lossy nature of spectral analysis of Markov
chains on exponential-sized state spaces. In our main technical result, we show
a general way of deriving entropy contraction, a.k.a. modified log-Sobolev
inequalities, for down-up random walks from spectral notions. We show that
spectral independence of a distribution under arbitrary external fields
automatically implies entropic independence.
  To derive our results, we relate entropic independence to properties of
polynomials: $\mu$ is entropically independent exactly when a transformed
version of the generating polynomial of $\mu$ is upper bounded by its linear
tangent; this property is implied by concavity of the said transformation,
which was shown by prior work to be locally equivalent to spectral
independence. We apply our results to obtain tight modified log-Sobolev
inequalities and mixing times for multi-step down-up walks on fractionally
log-concave distributions. As our flagship application, we establish the tight
mixing time of $O(n\log n)$ for Glauber dynamics on Ising models whose
interaction matrix has eigenspectrum lying within an interval of length smaller
than $1$, improving upon the prior quadratic dependence on $n$.
