  We introduce a Laplacian separation principle for the the eikonal equation on
Markov chains. As application, we prove an isoperimetric concentration
inequality for Markov chains with non-negative Ollivier curvature. That is,
every single point from the concentration profile yields an estimate for every
point of the isoperimetric estimate. Applying to exponential and Gaussian
concentration, we obtain affirmative answers to two open quesions by Erbar and
Fathi. Moreover, we prove that the modified log-Sobolev constant is at least
the minimal Ollivier Ricci curvature, assuming non-negative Ollivier sectional
curvature, i.e., the Ollivier Ricci curvature when replacing the $\ell_1$ by
the $\ell_\infty$ Wasserstein distance. This settles a recent open Problem by
Pedrotti. We give a simple example showing that non-negative Ollivier sectional
curvature is necessary to obtain a modified log-Sobolev inequality via positive
Ollivier Ricci bound. This provides a counterexample to a conjecture by Peres
and Tetali.
