The annealed free energy of the output distribution from a rate-R random code decomposes into bulk and sparse branches whose competition yields an explicit phase diagram in the (β, R) plane meeting at the mutual information point.
Exact channel resolvability exponents for the soft-covering problem,
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A Statistical-Physics Refinement of Soft Covering
The annealed free energy of the output distribution from a rate-R random code decomposes into bulk and sparse branches whose competition yields an explicit phase diagram in the (β, R) plane meeting at the mutual information point.