2021
DOI: 10.48550/arxiv.2106.04105
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Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models

Abstract: We introduce a notion called entropic independence for distributions µ defined on pure simplicial complexes, i.e., subsets of size k of a ground set of elements. Informally, we call a background measure µ entropically independent if for any (possibly randomly chosen) set S, the relative entropy of an 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 spectral independe… Show more

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Cited by 12 publications
(46 citation statements)
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“…The next proposition shows that spectral domination of µ implies a restricted form of entropic independence (see [Ana+21]), for probability distributions ν which are O(1)-bounded with respect to µ. In the special case that we can take ǫ = ∞, we do not need the O(1)-boundedness condition, and recover the fact that fractional log concavity implies entropic independence from [Ana+21].…”
Section: Restricted Entropy Contraction For Field Dynamicsmentioning
confidence: 78%
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“…The next proposition shows that spectral domination of µ implies a restricted form of entropic independence (see [Ana+21]), for probability distributions ν which are O(1)-bounded with respect to µ. In the special case that we can take ǫ = ∞, we do not need the O(1)-boundedness condition, and recover the fact that fractional log concavity implies entropic independence from [Ana+21].…”
Section: Restricted Entropy Contraction For Field Dynamicsmentioning
confidence: 78%
“…In this work, we develop new methods for analyzing the mixing time of Markov chains and as an application, finally prove that a slight variation of Glauber dynamics, which we dub balanced Glauber dynamics, mixes in the optimal O(n log n) many steps up to the appropriate uniqueness thresholds for both the hardcore and Ising models. Our results build upon tools from recent works which analyze Markov chains through high-dimensional expansion properties [e.g., ALO21; CLV21; Che+21] and in particular, techniques for establishing entropic independence introduced by Anari, Jain, Koehler, Pham, and Vuong [Ana+21]. To surpass the limitations of previous methods, we introduce a number of new ideas for proving and using functional inequalities restricted to certain classes of well-behaved functionals.…”
Section: Introductionmentioning
confidence: 95%
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