Directed Polymers in Random Media and Spin Glass Models on Trees

  • F. Koukiou
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 84)


Using some results of the theory of branching random walks, we give a unifying framework for the mean-field theory for models of spin glasses and directed polymers in a random medium defined on regular trees. Their phase diagram is studied in the complex plane of temperature.


Spin Glass Random Medium Complex Weight Directed Polymer High Temperature Behaviour 
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  1. [1]
    Asmussen, S., Hering, H.: Branching processes, Birkhiuser, Basel (1983). zbMATHGoogle Scholar
  2. [2]
    Athreya, K. B., Ney, P. E.: Branching processes, Springer-Verlag, Berlin (1972). CrossRefzbMATHGoogle Scholar
  3. [3]
    Biggins, J.D.: Uniform convergence of martingales in the branching random walk. Ann. Prob. 20 137–151 (1992). MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Biggins, J.D.: Martingale convergence in the branching random walk. J. Appl. Probab. 14 25–37 (1977). MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Buffet, E., Patrick, A., Pulé, J.V.: Directed polymers on trees: a martingale approach. Preprint DIAS STP 91–34 (1991). Google Scholar
  6. [6]
    Chauvin, B., Rouault, A.: Boltzmann-Gibbs weights in the branching random walk. These proceedings.Google Scholar
  7. [7]
    Chayes, J.T., Chayes, L., Sethna, J.P., Thouless, D.J.: A mean field spin glass with short-range interactions. Commun. Math. Phys. 106 41–89 (1986). MathSciNetCrossRefGoogle Scholar
  8. [8]
    Collet, P., Koukiou, F.: Large deviations for multiplicative chaos. Commun. Math. Phys. 147 329–342 (1992). MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Derrida, B.: Random-energy model: an exactly solvable model of disordered systems. Phys. Rev. B24, 2613–2626 (1981). MathSciNetGoogle Scholar
  10. [10]
    Derrida, B.: A generalization of the random energy model which includes correlations between energies. J. Phys. Lett. 46 L401–407 (1985). MathSciNetCrossRefGoogle Scholar
  11. [11]
    Derrida, B.: Directed polymers in a random medium. Physica A163 71–84 (1990). MathSciNetGoogle Scholar
  12. [12]
    Derrida, B., Evans, M.R., Speer, E.R.: Mean field theory of directed polymers with random complex weights. Commun. Math. Phys. 156 221–244 (1993). MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Derrida, B., Spohn, H.: Polymers on Disordered trees, spin glasses, and traveling waves. J. Stat. Phys. 51 817–840 (1988). MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Edwards, S.F., Anderson, P.W.: Theory of spin glasses. J. Phys. F5, 965–974 (1975). CrossRefGoogle Scholar
  15. [15]
    Grimmett, G.: Percolation. Springer-Verlag, Berlin (1989). zbMATHGoogle Scholar
  16. [16]
    Holley, R., Waymire, E.C.: Multifractal dimensions and scaling exponents for strongly bounded random cascades. Ann. Appl. Prob. 2 819–845 (1992). MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Kahane, J.-P., Peyrière, J.: Sur certaines martingales de Benoit Mandelbrot. Adv. in Math. 22 131–145 (1976). CrossRefzbMATHGoogle Scholar
  18. [18]
    Koukiou, F.: A random covering interpretation for the phase transition of the random energy model. J. Stat. Phys. 60, 669–674 (1990). MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Koukiou, F.: Rigorous bounds for the free energy of the short-range Ising spin glass model. Europhys. Lett.17 669–671 (1992). MathSciNetGoogle Scholar
  20. [20]
    Koukiou, F.: The spin glass model on diamond lattices. J. Phys. A 28 2737–2743 (1995). MathSciNetGoogle Scholar
  21. [21]
    Koukiou, F., Picco, P.: Poisson point processes, cascades and random coverings of Rn . J. Stat. Phys. 62 481-489 (1991). MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Neveu, J.: Discrete-parameter martingales. Amsterdam- North-Holland 1975. (1986). Google Scholar
  23. [23]
    Parisi, G.: On the replica approach to random directed polymers in two dimension. J. Physique 51 1595–1606 (1990). MathSciNetCrossRefGoogle Scholar
  24. [24]
    Ruelle, D.: Thermodynamic formalism. Addison-Wesley (1978).Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • F. Koukiou
    • 1
    • 2
  1. 1.Groupe de Physique StatistiqueUniversité de Cergy-PontoiseCergy-Pontoise CedexFrance
  2. 2.Centre de Physique ThéoriqueEcole PolytechniquePalaiseau CedexFrance

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