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One-dimensional disordered Ising models by replica and cavity methods.

C Lucibello1, F Morone1, T Rizzo2

  • 1Dipartimento di Fisica, Università "La Sapienza," P.le A. Moro 2, I-00185 Rome, Italy.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 15, 2014
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This summary is machine-generated.

This study introduces a novel spectral decomposition method for disordered Ising models, yielding exact analytical expressions for correlation functions and free energies in various systems. The findings offer efficient numerical approximations and rigorous derivations without replicas.

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Area of Science:

  • Statistical mechanics
  • Disordered systems
  • Spin models

Background:

  • Disordered Ising models present complex behavior.
  • Understanding correlation functions and free energies is crucial.
  • Replicated transfer matrix methods are established but can be complex.

Purpose of the Study:

  • To develop a formalism for analyzing disordered Ising models.
  • To derive exact analytical expressions for correlation functions and free energies.
  • To provide efficient numerical approximations for these quantities.

Main Methods:

  • Spectral decomposition of the replicated transfer matrix.
  • Analysis of isolated one-dimensional systems.
  • Application to locally treelike graph and factor graph (p-spin) ensembles.
  • Probabilistic approach with cavity method (for replica-free derivations).

Main Results:

  • Exact analytical expressions for correlation functions and average free energies.
  • Efficient numerical approximations for open and closed finite chains.
  • Rigorous replica-free derivations for most results using a cavity-method-like approach.

Conclusions:

  • The spectral decomposition formalism provides a powerful tool for disordered Ising models.
  • The derived expressions and methods are applicable to diverse systems.
  • The study offers both analytical insights and practical numerical approximations.