{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T06:01:44Z","timestamp":1781157704898,"version":"3.54.1"},"reference-count":14,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2026,6,8]],"date-time":"2026-06-08T00:00:00Z","timestamp":1780876800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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(\u03b2|C)=\u2211yn[PYn|C(yn)]\u03b2, where yn is the channel output vector, C is the code, and \u03b2&gt;0 plays the role of inverse temperature. More precisely, our focus is on the associated annealed free energy, \u03c8(\u03b2,R)=limn\u2192\u221e1nlogE[Zn(\u03b2|C)], where the expectation is with respect to the randomness of C. This quantity encodes the full R\u00e9nyi spectrum of the output distribution. The single-letter formula derived for the annealed free energy decomposes into two branches, which reflect a \u201ccompetition\u201d between two populations of codewords. One is the bulk branch, \u03c8b(\u03b2,R), which is driven by typical codewords, and the other one is the sparse branch\u03c8s(\u03b2,R), which is driven by atypical codewords, where the qualifiers \u2018typical\u2019 and \u2018atypical\u2019 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 \u03b2&gt;0. The phase boundary R\ud83d\udfc9(\u03b2), where the two branches are equal, is analyzed for \u03b2\u22651, where it has an explicit closed-form expression. The phase diagram in the first quadrant of the (\u03b2,R) plane has four regions separated by three boundaries: R=Ib(\u03b2) (bulk branch transition), R=R\ud83d\udfc9(\u03b2) (bulk\u2013sparse competition boundary), and R=Is(\u03b2) (sparse branch transition), all meeting at the point (\u03b2,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.<\/jats:p>","DOI":"10.3390\/e28060647","type":"journal-article","created":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T00:43:54Z","timestamp":1780965834000},"page":"647","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Statistical-Physics Refinement of Soft Covering"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9547-3243","authenticated-orcid":false,"given":"Neri","family":"Merhav","sequence":"first","affiliation":[{"name":"The Viterbi Faculty of Electrical and Computer Engineering, Technion\u2014Israel Institute of Technology, Technion City, Haifa 3200003, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,6,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"752","DOI":"10.1109\/18.256486","article-title":"Approximation theory of output statistics","volume":"39","author":"Han","year":"1993","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Cuff, P. (2016, January 10\u201315). Soft covering with high probability. Proceedings of the 2016 IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain.","DOI":"10.1109\/ISIT.2016.7541842"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1562","DOI":"10.1109\/TIT.2006.871040","article-title":"General nonasymptotic and asymptotic formulas in channel resolvability and identification capacity and their application to wire-tap channel","volume":"52","author":"Hayashi","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Hou, J., and Kramer, G. (July, January 29). Effective secrecy: Reliability, confusion and stealth. 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(2011). Information Theory: Coding Theorems for Discrete Memoryless Systems, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9780511921889"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2505","DOI":"10.1109\/18.720546","article-title":"The method of types","volume":"44","year":"1998","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1561\/0100000142","article-title":"A toolbox for refined information-theoretic analyses with applications","volume":"22","author":"Merhav","year":"2025","journal-title":"Found. Trends Commun. Inf. Theory"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/6\/647\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T05:52:49Z","timestamp":1781157169000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/6\/647"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,8]]},"references-count":14,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2026,6]]}},"alternative-id":["e28060647"],"URL":"https:\/\/doi.org\/10.3390\/e28060647","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,8]]}}}