Modified log-Sobolev inequalities for strong-Rayleigh measures

We establish universal modified log-Sobolev inequalities for reversible Markov chains on the boolean lattice $\{0,1\}^n$, under the only assumption that the invariant law $\pi$ satisfies a form of negative dependence known as the \emph{stochastic covering property}. This condition is strictly weaker than the strong Rayleigh property, and is satisfied in particular by all determinantal measures, as well as any product measure over the set of bases of a balanced matroid. In the special case where $\pi$ is $k-$homogeneous, our results imply the celebrated concentration inequality for Lipschitz functions due to Pemantle \& Peres (2014). As another application, we deduce that the natural Monte-Carlo Markov Chain used to sample from $\pi$ has mixing time at most $kn\log\log\frac{1}{\pi(x)}$ when initialized in state $x$. This considerably improves upon the $kn\log\frac{1}{\pi(x)}$ estimate recently obtained under stronger assumptions by Anari, Oveis Gharan \& Rezaei (2016).