Transport-entropy inequalities and curvature in discrete-space Markov   chains
  We show that if the random walk on a graph has positive coarse Ricci curvature in the sense of Ollivier, then the stationary measure satisfies a W^1 transport-entropy inequality. Peres and Tetali have conjectured a stronger consequence, that a modified log-Sobolev inequality (MLSI) should hold, in analogy with the setting of Markov diffusions. We discuss how our entropy interpolation approach suggests a natural attack on the MLSI conjecture. 
Ronen Eldan and James R. Lee and Joseph Lehec
math.PR math.MG