High-Dimensional Ultra-Log-Concave Distributions

Source abstract

Ultra-log-concave distributions are ubiquitous in probability, combinatorics, and statistical mechanics and have been studied extensively. In this paper, we introduce a quantitative high-dimensional extension of this notion, called $δ$-ultra-log-concavity, for probability measures on $\mathbb{N}^d$ with downward closed support. When $δ= 1$, this notion coincides with the class studied by Gurvits (2009) via strongly log-concave generating functions, and with the class defined by Anari, Oveis Gharan, and Vinzant (2021) via completely log-concave generating functions; in one dimension, it reduces to classical ultra-log-concavity. We establish several functional inequalities, including a weighted Poincaré inequality, a discrete Brascamp--Lieb inequality, and a weighted Wu-type modified log-Sobolev inequality. Our approach combines integrated Bakry--Émery calculus for a canonical birth-death chain with Poisson stochastic localization, which arises as the time reversal of coordinatewise binomial thinning. We further establish concentration of measure, maximum-entropy principles, and several closure properties for ultra-log-concave measures, and develop applications to queueing models, polymatroids, antiferromagnetic Potts models, and hardcore models. Finally, a lattice scaling limit of the discrete theory yields Poincaré and Brascamp--Lieb inequalities for Laguerre diffusions.