Communications in Mathematical Physics

, Volume 156, Issue 2, pp 221–244 | Cite as

Mean field theory of directed polymers with random complex weights

  • B. Derrida
  • M. R. Evans
  • E. R. Speer


We show that for the problem of directed polymers on a tree with i.i.d. random complex weights on each bond, three possible phases can exist; the phase of a particular system is determined by the distribution ρ of the random weights. For each of these three phases, we give the expression of the free energy per unit length in the limit of infinitely long polymers. Our proofs require several hypotheses on the distribution ρ, most importantly, that the amplitude and the phase of each complex weight be statistically independent. The main steps of our proofs use bounds on noninteger moments of the partition function and self averaging properties of the free energy. We illustrate our results by some examples and discuss possible generalizations to a larger class of distributions, to Random Energy Models, and to the finite dimensional case. We note that our results are not in agreement with the predictions of a recent replica approach to a similar problem.


Polymer Neural Network Free Energy Field Theory Complex System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • B. Derrida
    • 1
  • M. R. Evans
    • 1
  • E. R. Speer
    • 2
  1. 1.Service de Physique ThéoriqueCEN SaclayGif-sur-YvetteFrance
  2. 2.School of MathematicsInstitute for Advanced StudyPrincetonUSA

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