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We introduce a notion called entropic independence that is an entropic analogof spectral notions of high-dimensional expansion. Informally, entropicindependence of a background distribution $\mu$ on $k$-sized subsets of aground set of elements says that for any (possibly randomly chosen) set $S$,the relative entropy of a single element of $S$ drawn uniformly at randomcarries at most $O(1/k)$ fraction of the relative entropy of $S$. Entropicindependence is the analog of the notion of spectral independence, if onereplaces variance by entropy. We use entropic independence to derive tightmixing time bounds, overcoming the lossy nature of spectral analysis of Markovchains on exponential-sized state spaces. In our main technical result, we showa general way of deriving entropy contraction, a.k.a. modified log-Sobolevinequalities, for down-up random walks from spectral notions. We show thatspectral independence of a distribution under arbitrary external fieldsautomatically implies entropic independence.  To derive our results, we relate entropic independence to properties ofpolynomials: $\mu$ is entropically independent exactly when a transformedversion of the generating polynomial of $\mu$ is upper bounded by its lineartangent; this property is implied by concavity of the said transformation,which was shown by prior work to be locally equivalent to spectralindependence. We apply our results to obtain tight modified log-Sobolevinequalities and mixing times for multi-step down-up walks on fractionallylog-concave distributions. As our flagship application, we establish the tightmixing time of $O(n\log n)$ for Glauber dynamics on Ising models whoseinteraction matrix has eigenspectrum lying within an interval of length smallerthan $1$, improving upon the prior quadratic dependence on $n$.

Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models