Sharp log-Sobolev inequalities on finite cyclic groups

Abstract

Let ${\mathbb{Z}}_n$ be the cyclic group equipped with the uniform probability measure $\ddot{I}€$, and let $A_{\ddot{I}{ˆ}_n}$ 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 the nearest-neighbor simple random walk. We use their cubic-majorant reduction, which turns the problem into a third-moment estimate; the new point is a blockwise third-moment estimate adapted to the word-length multiplier. The same third-moment argument also recovers the log-Sobolev inequality for the Poisson semigroup on the circle, first proved by Weissler~\cite{Weissler1980}. The same sharp inequalities were independently obtained by Yao~\cite{Yao2026} using a different method.

10 pages. Minor revision. Acknowledgement (AI tools) added. Lean formalization available at https://github.com/XinyuanXie-hub/cyclic-lsi-new