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

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 μ 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, a constant multiple of its “share of entropy.” Entropic independence is the natural analog of the recently established notion of spectral independence, if one replaces variance by entropy. We use entropic independence to derive tight mixing time bounds for natural random walks associated with a number of widely studied distributions, 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 much simpler spectral notions. We show that spectral independence of a distribution under arbitrary external fields automatically implies entropic independence. Our result can be seen as a new framework to establish entropy contraction from the much simpler variance contraction inequalities. To derive our results, we relate entropic independence to properties of polynomials: μ is entropically independent exactly when a transformed version of the generating polynomial of μ 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. Our framework makes no assumptions on marginals of μ or the degrees of the underlying graphical model when μ is based on one, and it has the ability to derive tight bounds on mixing time even when it is not nearly-linear. 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.