Intrinsic regularity in the discrete log-Sobolev inequality

The chain rule lies at the heart of the powerful Gamma calculus for Markov diffusions on manifolds, providing remarkable connections between several fundamental notions such as Bakry-\'Emery curvature, entropy decay, and hypercontractivity. For Markov chains on finite state spaces, approximate versions of this chain rule have recently been put forward, with an extra cost that depends on the log-Lipschitz regularity of the considered observable. Motivated by those findings, we here investigate the regularity of extremizers in the discrete log-Sobolev inequality. Specifically, we show that their log-Lipschitz constant is bounded by a universal multiple of $\log d$, where $d$ denotes the inverse of the smallest non-zero transition probability. As a consequence, we deduce that the log-Sobolev constant of any reversible Markov chain on a finite state space is at least a universal multiple of $\kappa/\log d$, where $\kappa$ is the Bakry-\'Emery curvature. This is a sharp discrete analogue of what is perhaps the most emblematic application of the Bakry-\'Emery theory for diffusions. We also obtain a very simple proof of the main result in \cite{MR4620718}, which asserts that the log-Sobolev constant and its modified version agree up to a $\log d$ factor. Our work consolidates the role of the sparsity parameter $\log d$ as a universal cost for transferring results from Markov diffusions to discrete chains.