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 \[
\psi_n(k) = \min(k,n-k). \] We prove the sharp log-Sobolev inequality \[
\text{Ent}_{\pi}(f^2)
\le 2\pi(f A_{\psi_n} f),
\qquad f:\mathbb Z_n \to [0,\infty), \] 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.
From: Haonan Zhang [view email]
[v1]
Mon, 1 Jun 2026 20:11:29 UTC (8 KB)
[v2]
Tue, 9 Jun 2026 18:35:54 UTC (10 KB)
[v3]
Sat, 1 Aug 2026 03:17:40 UTC (9 KB)
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