Dispersion Bound for the Wyner-Ahlswede-Körner Network via Reverse Hypercontractivity on Types

Using the functional-entropic duality and the reverse hypercontractivity of the transposition semigroup, we lower bound the error probability for each joint type in the Wyner-Ahlswede-Korner problem. Then by averaging the error probability over types, we lower bound the c-dispersion (which characterizes the second-order behavior of the weighted sum of the rates of the two compressors when a nonvanishing error probability is small) as the variance of the gradient of $\inf_{P_{U\vert X}}\{cH(Y\vert U)+ \ I(U;X)\}$ with respect to $Q_{XY}$, the per-letter side information and source distribution. On the other hand, using the method of types we derive a new upper bound on the c-dispersion, which improves the existing upper bounds but has a gap to the aforementioned lower bound.