Abstract
This is a short review of systems which exhibit non-self-averaging effects: sums of random variables when the distribution has a long tail, mean field spin glasses, random map models and returns of a random walk to the origin. We will see that the non- self-averaging effects are identical in the case of sums of random variables and in the spin glass problem as predicted by the replica approach. Also we will see that for the random map models or for the problem of the returns of a random walk to the origin, the non-self-averaging effects coincide with the results of the replica approach when the number n of replica equals -1/2 or - 1.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
M. Mézard, G. Parisi, N. Sourlas, G. Toulouse and M. Virasoro, Replica symmetry breaking and the nature of the spin glass phase, J. Physique ,45, 843 (1984).
M. Mézard, G. Parisi and M. Virasoro, “Spin Glass Theory and Beyond”, World Scientific (1987).
J.P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mecha nisms, models and physical applications, Phys. Rep. ,195, 127 (1990).
G. Parisi, The order parameter for spin glasses: A function on the interval 0-1, J. Phys. A ,48, 1101(1980).
D. Sherrington and S. Kirkpatrick, Solvable model of a spin glass, Phys. Rev. Lett. ,35, 1792 (1975).
S. Kirkpatrick and D. Sherrington, Infinite ranged models of spin glasses, Phys. Rev. ,B17, 4385 (1978).
B. Derrida, Random energy model, an exactly solvable model of disordered systems, Phys. Rev. ,B24, 2613 (1981).
B. Derrida and E. Gardner, Magnetic properties and the function q(x) of the generalised random energy model, J. Phys. C ,19, 5783 (1986).
B. Derrida, Directed polymers in a random medium, Physica ,163, 71 (1990).
J.L. van Hemmen and R.G. Palmer, The replica method and a solvable spin glass model, J. Phys. A ,12, 563 (1979).
D. Sherrington, Ising replica magnet, J. Phys. A ,13, 637 (1980).
A. Blandin, Theories versus experiments in the spin glass systems, J. Physique ,C6, 1499 (1978).
J.R.L. de Almeida and D.J. Thouless, Stability of the Sherrington-Kirkpatrick solution of a spin glass model, J. Phys. A ,11, 983 (1978).
G. Parisi, Order parameter for spin glasses, Phys. Rev. Lett ,50, 1946 (1983).
B. Derrida and G. Toulouse, Sample to sample fluctuations in the random energy model, J. Physique Lett ,46, L223 (1985).
M. Mézard, G. Parisi and M.A. Virasoro, Random free energies in spin glasses, J. Physique Lett ,46, L217 (1985).
E. Gardner and B. Derrida, The probability distribution of the partition function of the random energy model, J. Phys. A ,22, 1975 (1989).
S.A. Kauffman, “The Origin of Order”, Oxford University Press (1993).
B. Derrida and H. Flyvbjerg, Multivalley structure in Kauffman’s model: Analogy with spin glasses, J. Phys. A ,19, L1003 (1986).
B. Derrida and H. Flyvbjerg, The random map model, a disordered model with deterministic dynamics, J Physique ,48, 971 (1987).
B. Derrida and H. Flyvbjerg, Statistical properties of randomly broken objects and of multivalley structures in disordered systems, J. Phys. A ,20, 5273 (1987).
I. Kondor, Parisi’s mean field solution for spin glasses as an analytic continuation in the replica number, J. Phys. A ,16, L127 (1993).
E. Buffet, A. Patrick and J.V. Pule, Directed polymers on a tree: A martingale approach, J. Phys. A ,26, 1823 (1983).
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, 3681 (1993).
B. Derrida and D. Bessis, Statistical properties of valleys in the annealed random map model, J. Phys. A ,21, L509 (1988).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Science+Business Media New York
About this chapter
Cite this chapter
Derrida, B. (1994). Non-Self-Averaging Effects in Sums of Random Variables, Spin Glasses, Random Maps and Random Walks. In: Fannes, M., Maes, C., Verbeure, A. (eds) On Three Levels. NATO ASI Series, vol 324. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2460-1_12
Download citation
DOI: https://doi.org/10.1007/978-1-4615-2460-1_12
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-6047-6
Online ISBN: 978-1-4615-2460-1
eBook Packages: Springer Book Archive