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This paper introduces a new converse machinery for a challenging class ofdistributed source-type problems (e.g.\ distributed source coding, commonrandomness generation, or hypothesis testing with communication constraints),through the example of the Wyner-Ahlswede-K\"orner network. Using thefunctional-entropic duality and the reverse hypercontractivity of thetransposition semigroup, we lower bound the error probability for each jointtype. Then by averaging the error probability over types, we lower bound the$c$-dispersion (which characterizes the second-order behavior of the weightedsum of the rates of the two compressors when a nonvanishing error probabilityis small) as the variance of the gradient of $\inf_{P_{U|X}}\{cH(Y|U)+I(U;X)\}$with respect to $Q_{XY}$, the per-letter side information and sourcedistribution. In comparison, using standard achievability arguments based onthe method of types, we upper-bound the $c$-dispersion as the variance of$c\imath_{Y|U}(Y|U)+\imath_{U;X}(U;X)$, which improves the existing upperbounds but has a gap to the aforementioned lower bound.

Dispersion Bound for the Wyner-Ahlswede-K\"orner Network via Reverse Hypercontractivity on Types