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A Statistical-Physics Refinement of Soft Covering.

Neri Merhav1

  • 1The Viterbi Faculty of Electrical and Computer Engineering, Technion-Israel Institute of Technology, Technion City, Haifa 3200003, Israel.

Entropy (Basel, Switzerland)
|June 26, 2026
PubMed
Summary
This summary is machine-generated.

We analyze random codes using statistical physics, revealing two distinct codeword behaviors: typical (bulk) and atypical (sparse). Their competition defines a phase diagram with implications for information theory applications.

Keywords:
Rényi entropyannealed free energychannel resolvabilityfree energyguessworkhypothesis testingphase transitionsrandom codingsoft coveringstatistical mechanics

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Area of Science:

  • Information Theory
  • Statistical Physics
  • Coding Theory

Background:

  • Studying channel output distributions is crucial for understanding information transmission.
  • Random codes are fundamental in coding theory, but their output distributions are complex.
  • Statistical physics offers powerful tools for analyzing complex systems, including random codes.

Purpose of the Study:

  • To analyze the channel output distribution induced by random codes using statistical physics.
  • To investigate the annealed free energy and its connection to the Rényi spectrum.
  • To characterize the competition between typical (bulk) and atypical (sparse) codewords.

Main Methods:

  • Utilized statistical physics concepts, including partition functions and annealed free energy.
  • Derived a single-letter formula for the annealed free energy, decomposing it into bulk and sparse branches.
  • Analyzed the phase structure of each branch and their competition, defining transition boundaries.

Main Results:

  • The annealed free energy encodes the full Rényi spectrum of the output distribution.
  • A phase diagram is established with four regions defined by bulk and sparse branch transitions.
  • The competition boundary R*(β) between bulk and sparse branches has a closed-form expression for β≥1.

Conclusions:

  • The study provides a novel statistical physics framework for analyzing random codes.
  • The identified bulk and sparse branches offer insights into codeword behavior and competition.
  • Results have potential applications in guesswork, channel resolvability, and hypothesis testing.