Sharp log-Sobolev inequalities 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 \[ ψ_n(k) = \min(k,n-k). \] We prove the sharp log-Sobolev inequality \[ \text{Ent}_π(f^2) \le 2π(f A_{ψ_n} f), \qquad f:\mathbb Z_n \to [0,\infty), \] for every $n \ge 4$. The proof is inspired by the recent work of Frank and Ivanisvili~\cite{FrankIvanisvili2026} on a sharp log-Sobolev inequality for nearest-neighbor simple random walk. We use their cubic-majorant reduction, which turns the problem into a 3rd moment estimate; the new point is a blockwise 3rd moment estimate adapted to the word-length multiplier. The same 3rd moment argument also recovers the log-Sobolev inequality for Poisson-semigroup on the circle, first proved by Weissler~\cite{Weissler1980}. The same sharp inequalities were also obtained recently by Yao~\cite{Yao2026} by a different method.