Displacement convexity of entropy and related inequalities on graphs

@article{Gozlan2012DisplacementCO,
  title={Displacement convexity of entropy and related inequalities on graphs},
  author={Nathael Gozlan and Cyril Roberto and Paul-Marie Samson and Prasad Tetali},
  journal={Probability Theory and Related Fields},
  year={2012},
  volume={160},
  pages={47-94}
}
We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path and derive Prékopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings. To illustrate through examples, we apply our results to the complete graph and to the hypercube for which our results are optimal… 
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