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 $π$, and let $A_{ψ_n}$ be the Laplacian with word length $$ ψ_n(k) = \min(k,n-k). $$ For every $n\ge4$, we prove the sharp log-Sobolev inequality $$ \text{Ent}_π(|f|^2) \le 2π(\bar{f}A_{ψ_n} f), \qquad f:\mathbb Z_n \to \mathbb{C}, $$ where $\text{Ent}_π$ 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 $n\ge 4$ and $f:\mathbb Z_n\to[0,\infty)$ with $π(f^2)=1$, $$ 2π(fA_{ψ_n}f)-\operatorname{Ent}_π(f^2) \ge \frac1{12}\|f-1\|_{L^2(π)}^4, $$ with coefficient $1/(12d)$ on products $(\mathbb Z_n)^d$. The quartic order and the $d^{-1}$ dependence are optimal.