Sharp log-Sobolev inequalities and quartic stability on finite cyclic groups

Abstract

Let ${\mathbb{Z}}_n$ be the cyclic group equipped with the uniform probability measure $\pi$, and let $A_{{\psi }_n}$ be the Laplacian with word length $${\psi }_n(k)=\min (k,n-k).$$ For every $n\ge 4$, we prove the sharp log-Sobolev inequality $${\text{Ent}}_{\pi }(|f{|}^2)\le 2\pi (\bar{f}{A}_{{\psi }_n}f),f:{\mathbb{Z}}_n\to \mathbb{C},$$ where ${\text{Ent}}_{\pi }$ is the relative entropy with respect to $\pi$. Equivalently, the Poisson semigroup $P_t=e^{-t{A}_{{\psi }_n}}$ 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 $n\ge 4$ and $f:{\mathbb{Z}}_n\to [0,\infty )$ with $\pi (f^2)=1$, $$2\pi (f{A}_{{\psi }_n}f)-{\mathrm{Ent}}_{\pi }(f^2)\ge \frac{1}{12}\Vert f-1{\Vert }_{L^2(\pi )}^4,$$ with coefficient $1/(12d)$ on products $({\mathbb{Z}}_n{)}^d$. The quartic order and the $d^{-1}$ dependence are optimal.

14 pages. Minor revision. Stability result added. Title changed