Entropic independence via sparse localization

TL;DR

This paper introduces sparse localization to establish entropic independence in measures, enabling proofs of entropy conservation in bounded degree graphs without requiring full spectral independence.

Contribution

It develops a restricted localization framework that requires independence only on sparse pinnings, broadening applicability in models where previous criteria fail.

Findings

Proves quadratic entropic stability under sparse localization.

Establishes entropic independence with explicit loss depending on sparsity.

Provides a rigorous proof of entropy conservation in bounded degree graphs.

Abstract

Entropic independence is a structural property of measures that underlies modern proofs of functional inequalities, notably (modified) log-Sobolev inequalities, via ``annealing'' or local-to-global schemes. Existing sufficient criteria for entropic independence typically require spectral independence and/or uniform bounds on marginals under \emph{all} pinnings, which can fail in natural canonical-ensemble models even when strong mixing properties are expected. We introduce \emph{sparse localization}: a restricted localization framework, in the spirit of Chen--Eldan, in which one assumes $\ell_2$-independence only for a sparse family of pinnings (those fixing at most $cn$ coordinates for any $c > 0$), yet still deduces quadratic entropic stability and entropic independence with an explicit multiplicative loss of order $c^{-1}$. As an application, we give a rigorous proof of approximate conservation of entropy for the uniform distribution on independent sets of a given size in bounded degree graphs.