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

Understand

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.