Intrinsic regularity in the discrete log-Sobolev inequality
Justin Salez, Pierre Youssef
TL;DR
This work provides a sharp discrete analogue of Bakry–Émery theory by establishing intrinsic regularity for extremizers of the discrete log-Sobolev inequality and leveraging it to relate the log-Sobolev constant to the modified version with a universal $\log d$-cost. The authors prove that any LSI extremizer satisfies $\mathrm{Lip}(\log f) \le 14\log d$, which yields $t_{ls} \le 15\log d\; t_{mls}$ and, for positive Bakry–Émery curvature $\kappa$, $t_{ls} \le (33\log d)/\kappa$, plus a concise proof of the analogous result in MR4620718. They also derive a discrete Bakry–Émery-type bound without requiring a full chain rule, using three key lemmas to control the semigroup regularity, Dirichlet forms, and an approximate chain rule, culminating in a universal $\log d$ factor reflecting the sparsity of the chain. The results emphasize the central role of the sparsity parameter $d$ as the universal cost for transferring diffusion-based insights to discrete chains, and they connect discrete LSI theory to curvature notions in Markov chains with broad implications for finite-state hypercontractivity and entropy decay.
Abstract
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-Émery 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 $κ/\log d$, where $κ$ is the Bakry-Émery curvature. This is a sharp discrete analogue of what is perhaps the most emblematic application of the Bakry-Émery 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.