A Statistical-Physics Refinement of Soft Covering

We study the channel output distribution induced by a random code of rate R from the perspective of statistical physics. The central object is the partition function Zn(β|C)=∑yn[PYn|C(yn)]β, where yn is the channel output vector, C is the code, and β>0 plays the role of inverse temperature. More precisely, our focus is on the associated annealed free energy, ψ(β,R)=limn→∞1nlogE[Zn(β|C)], where the expectation is with respect to the randomness of C. This quantity encodes the full Rényi spectrum of the output distribution. The single-letter formula derived for the annealed free energy decomposes into two branches, which reflect a "competition" between two populations of codewords. One is the bulk branch, ψb(β,R), which is driven by typical codewords, and the other one is the sparse branchψs(β,R), which is driven by atypical codewords, where the qualifiers 'typical' and 'atypical' are in a sense that will become apparent later. We analyze the phase structure of each branch separately and characterize their competition. Both branches are derived for all β>0. The phase boundary R🟉(β), where the two branches are equal, is analyzed for β≥1, where it has an explicit closed-form expression. The phase diagram in the first quadrant of the (β,R) plane has four regions separated by three boundaries: R=Ib(β) (bulk branch transition), R=R🟉(β) (bulk-sparse competition boundary), and R=Is(β) (sparse branch transition), all meeting at the point (β,R)=(1,I(X;Y)), where I(X;Y) is the mutual information induced by the input type and the channel. Applications to guesswork, channel resolvability, and hypothesis testing are discussed, and all the results are illustrated with a numerical example of a Z-channel.