Sharp log-Sobolev inequalities and quartic stability on finite cyclic groups
Abstract
Let ℤn be the cyclic group equipped with the uniform probability measure π, and let Aψn be the Laplacian with word length ψn(k) = min(k,n−k). For every n≥4, we prove the sharp log-Sobolev inequality Entπ(|f|2) ≤ 2π(f̅Aψn f), f:ℤn → ℂ, where Entπ is the relative entropy with respect to π. Equivalently, the Poisson semigroup Pt=e−tAψn satisfies the optimal hypercontractivity. The proof is inspired by the recent work of Frank and Ivanisvili [FI26] on a sharp log-Sobolev inequality for the nearest-neighbor simple random walk. Similar arguments yield a simple proof of Weissler's sharp log-Sobolev inequality for the Poisson semigroup on the circle \cite{Weissler1980}. The same inequalities were independently obtained by Yao~\cite{Yao2026} using a different method. We also prove quantitative stability estimates. For n≥ 4 and f:ℤn→[0,∞) with π(f2)=1, 2π(fAψnf)−Entπ(f2) ≥ 1/12|f−1|L2(π)4, with coefficient 1/(12d) on products (ℤn)d. The quartic order and the d−1 dependence are optimal.
Related
- Stability under mixtures of transportation inequalities and restricted log-Sobolev inequalities
- Stability of spinorial Sobolev inequalities on Sn
- Sharp uniform stability of the logarithmic Sobolev inequality on finite cycles and stability of the cubic Sobolev inequality
- A Note on Fractional Sobolev inequalities revisited
- Sharp Sobolev inequalities on minimal immersions via optimal transport
BibTeX
@misc{xie2026sharp,
title = {{Sharp log-Sobolev inequalities and quartic stability on finite cyclic groups}},
author = {Xinyuan Xie and Haonan Zhang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2606.02847},
eprint = {2606.02847},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.02847},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Sharp log-Sobolev inequalities and quartic stability on finite cyclic groups
AU - Xinyuan Xie
AU - Haonan Zhang
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2606.02847
UR - https://arxiv.org/abs/2606.02847
AB - Let ℤ_n be the cyclic group equipped with the uniform probability measure π, and let A_ψ_n be the Laplacian with word length ψ_n(k) = min(k,n−k). For every n≥4, we prove the sharp log-Sobolev inequality Ent_π(|f|^2) ≤ 2π(f̅A_ψ_n f), f:ℤ_n → ℂ, where Ent_π is the relative entropy with respect to π. Equivalently, the Poisson semigroup P_t=e^−tA_ψ_n satisfies the optimal hypercontractivity. The proof is inspired by the recent work of Frank and Ivanisvili [FI26] on a sharp log-Sobolev inequality for the nearest-neighbor simple random walk. Similar arguments yield a simple proof of Weissler's sharp log-Sobolev inequality for the Poisson semigroup on the circle \cite{Weissler1980}. The same inequalities were independently obtained by Yao~\cite{Yao2026} using a different method. We also prove quantitative stability estimates. For n≥ 4 and f:ℤ_n→[0,∞) with π(f^2)=1, 2π(fA_ψ_nf)−Ent_π(f^2) ≥ 1/12|f−1|_L^2(π)^4, with coefficient 1/(12d) on products (ℤ_n)^d. The quartic order and the d^−1 dependence are optimal.
ER -