High-Dimensional Ultra-Log-Concave Distributions
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 ℕd 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.
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BibTeX
@misc{chen2026highdimensional,
title = {{High-Dimensional Ultra-Log-Concave Distributions}},
author = {Zongchen Chen and Sihan Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23994},
eprint = {2609.23994},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23994},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - High-Dimensional Ultra-Log-Concave Distributions
AU - Zongchen Chen
AU - Sihan Wang
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.23994
UR - https://arxiv.org/abs/2609.23994
AB - 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 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.
ER -