From random walks to spin glasses

Physica D: Nonlinear Phenomena

Volumn 107, Issue 2-4, 1997, Pages 186-198

  Derrida, B ^a  

^a ECOLE NORMALE SUPÉRIEURE   (France)

Author keywords

Broken symmetry of replica; Levy distributions; Non self averaging effects; Random walks; Spin glass

Indexed keywords

EID: 0008874438     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(97)00086-9     Document Type: Article

References (35)

1. M. Mezard, G. Parisi, N. Sourlas, G. Toulouse and M. Virasoro, Replica symmetry breaking and the nature of the spin glass phase, J. Physique 45 (1984) 843-854.
2. M. Mezard, G. Parisi and M. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).
3. J.P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep. 195 (1990) 127-293.
4. G. Parisi, The order parameter for spin glasses: A function on the interval 0-1, J. Phys. A 48 (1980) 1101-1112.
5. D. Sherrington and S. Kirkpatrick, Solvable model of a spin glass, Phys. Rev. Lett. 35 (1975) 1792-1796.
6. S. Kirkpatrick and D. Sherrington, Infinite ranged models of spin glasses, Phys. Rev. B 17 (1978) 4385-4403.
7. B. Derrida, Random energy model, an exactly solvable model of disordered systems, Phys. Rev. B 24 (1981) 2613-2626.
8. B. Derrida and E. Gardner, Magnetic properties and the function q(x) of the generalised random energy model, J. Phys. C 19 (1986) 5783-5798.
9. B. Derrida, Directed polymers in a random medium, Physica 163 (1980) 71-84.
10. J.L. van Hemmen and R.G. Palmer, The replica method and a solvable spin glass model, J. Phys. A 12 (1979) 563-580.
11. D. Sherrington, Ising replica magnet, J. Phys. A 13 (1980) 637-649.
12. A. Blandin, Theories versus experiments in the spin glass systems, J. Physique C 6 (1978) 1499-1516.
13. J.R.L. de Almeida and D.J. Thouless, Stability of the Sherrington-Kirkpatrick solution of a spin glass model, J. Phys. A 11 (1978) 983-990.
14. G. Parisi, Order parameter for spin glasses, Phys. Rev. Lett. 50 (1983) 1946-1948.
15. B. Derrida and G. Toulouse, Sample to sample fluctuations in the random energy model, J. Physique Lett. 46 (1985) L223-L228.
16. M. Mezard, G. Parisi and M.A. Virasoro, Random free energies in spin glasses, J. Physique Lett. 46 (1985) L217-L222.
17. E. Gardner and B. Derrida, The probability distribution of the partition function of the random energy model, J. Phys. A 22 (1989) 1975-1981.
18. S.A. Kauffman, The Origin of Order (Oxford University Press, Oxford, 1993).
19. B. Derrida and H. Flyvbjerg, Multivalley structure in Kauffman's model: Analogy with spin glasses, J. Phys. A 19 (1986) L1003-L1008.
20. B. Derrida and H. Flyvbjerg, The random map model, a disordered model with deterministic dynamics, J. Physique 48 (1987) 971-978.
21. B. Derrida and H. Flyvbjerg, Statistical properties of randomly broken objects and of multivalley structures in disordered systems, J. Phys. A 20 (1987) 5273-5288.
22. I. Kondor, Parisi's mean field solution for spin glasses as an analytic continuation in the replica number, J. Phys. A 16 (1993) L127-L131.
23. E. Buffet, A. Patrick and J.V. Pule, Directed polymers on a tree: A martingale approach, J. Phys. A 26 (1983) 1823-1834.
24. R.W. Penney, A.C. Coolen and D. Sherrington, Coupled dynamics of fast spins and slow interactions in neural networks and spin systems, J. Phys. A 26 (1993) 3681-3695.
25. B. Derrida and D. Bessis, Statistical properties of valleys in the annealed random map model, J. Phys. A 21 (1988) L509-L515.
26. B. Derrida and H. Flyvbjerg, unpublished.
27. J. Pitman and M. Yor, Two studies of Poisson-Dirichlet distributions preprint 1995, to appear.
28. J. Pitman and M. Yor, Random discrete distributions derived from self-similar random sets preprint 1995, to appear.
29. A.M. Vershik, The asymptotic distribution of factorizations of natural numbers into prime divisors, Soviet Math. Dokl. 34 (1986) 57-61.
30. A.M. Vershik and A.A. Shmidt, Limit measures arising in the theory of groups I, Theor. Prob. Appl. 22 (1977) 79-85.
31. A.M. Vershik and A.A. Shmidt, Limit measures arising in the theory of groups II, Theor. Prob. Appl. 23 (1978) 36-49.
32. P.G. Higgs, Frequency distributions in population genetics parallel those in statistical physics, Phys. Rev. E 51 (1995) 95-101.
33. G. Parisi, On the branching structure of the tree of states in spin glasses J. Stat. Phys. 72 (1993) 857-878.
34. D. Ruelle, A mathematical reformulation of Derrida's REM and GREM, Comm. Math. Phys. 108 (1987) 225-239.
35. B. Derrida, Non-self-averaging effects in sums of random variables, spin glasses, random maps and random walks, in: On Three Levels: Micro, Meso and Macroscopic Approaches in Physics, eds. M. Fannes et al. (Plenum Press, New York, 1994) pp. 125-137.