Displacement Convexity of Entropy on Graphs
summary
Defines Mt(A,B) as the set of graph midpoints and builds an explicit interpolation μ_t^π by choosing a random geodesic + binomial sampling with number of trials = d(x,y) and success p=t. Proves displacement convexity of entropy on the discrete cube for all t by using a Catalin Maron Tilda W2 that controls W1 (earth‑mover) and tensorization from two‑point spaces, recovers the Villani-style bound (half‑case previously had constant 1/16 scaling like 1/n), and leaves the symmetric group case mostly open (only t=1/2 is handled).
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