A substantial body of work has been devoted to finding analogs of positive Ricci curvature in discrete spaces. Earlier, the MCMC community discovered that path coupling is a powerful technique in establishing rapid mixing. In an ideal world, these two research threads would eventually unite, enriching both areas. A conjecture of Peres and Tetali offers a path to this resolution, but remains open.
Based on joint work with R. Eldan and J. Lehec, I will show how entropic interpolation can be used to establish an entropy-transport inequality in chains that admit a contracting coupling. By work of Bobkov and Gotze, this is a necessary condition for establishing a log-Sobolev inequality (and thus resolving the Peres-Tetali conjecture).