Spin Glasses: A New Direction for Probability Theory?

Michel Talagrand

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2000)

  • Volume: 11, Issue: S1, page 127-146
  • ISSN: 1120-6330

Abstract

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Physicists have proposed a wealth of new ideas concerning the collective behavior of large families of random variables, in which they have discovered the emergence of new types of order. The quest for mathematically rigorous proofs of their conjectures is likely to challenge and stimulate probabilists for years to come.

How to cite

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Talagrand, Michel. "Spin Glasses: A New Direction for Probability Theory?." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 11.S1 (2000): 127-146. <http://eudml.org/doc/289708>.

@article{Talagrand2000,
abstract = {Physicists have proposed a wealth of new ideas concerning the collective behavior of large families of random variables, in which they have discovered the emergence of new types of order. The quest for mathematically rigorous proofs of their conjectures is likely to challenge and stimulate probabilists for years to come.},
author = {Talagrand, Michel},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
language = {eng},
month = {12},
number = {S1},
pages = {127-146},
publisher = {Accademia Nazionale dei Lincei},
title = {Spin Glasses: A New Direction for Probability Theory?},
url = {http://eudml.org/doc/289708},
volume = {11},
year = {2000},
}

TY - JOUR
AU - Talagrand, Michel
TI - Spin Glasses: A New Direction for Probability Theory?
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2000/12//
PB - Accademia Nazionale dei Lincei
VL - 11
IS - S1
SP - 127
EP - 146
AB - Physicists have proposed a wealth of new ideas concerning the collective behavior of large families of random variables, in which they have discovered the emergence of new types of order. The quest for mathematically rigorous proofs of their conjectures is likely to challenge and stimulate probabilists for years to come.
LA - eng
UR - http://eudml.org/doc/289708
ER -

References

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  1. ALDOUS, D.J., Asymptotics in the random assignment problem. Probab. Th. ReL. Fields, 93, 1992, 507-534. Zbl0767.60006
  2. AIZEMAN, M. - LEBOWITZ, J. L. - RUELLE, D., Some rigorous results on the Sherrington-Kirkpatrick model. Commun Math. Phys., 112, 1987, 3-20. Zbl1108.82312
  3. AMIT, D. J. - GUTFREUND, H. - SOMPOLINSKY, H., Statistical mechanics of neural networks near saturation. Annals of Physics, 173, 1987, 30-67. 
  4. BOVIER, A. - GAYRARD, V., Hopfield models as generalized random meanfield models. In: A. Bovier - P. Picco (eds.), Mathematical aspects of spin glasses and neural networks. Progress in Probability, vol. 41, Birkhäuser, Boston1997. Zbl0910.60086
  5. BOUCHAUD, J.-P. - MÉZARD, M., Universality classes for extreme values statistics. J. Phys. A. Math. Gen., 30, 1997, 7997-8015. Zbl0953.60040
  6. A. BOVIER - P. PICCO (eds.), Mathematical aspects of spin glasses and neural networks. Progress in Probability, vol. 41, Birkhäuser, Boston1997. 
  7. COMETS, F. - NEVEU, J., Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: the high temperature case. Commun Math. Phys., 166, 1995, 549-564. Zbl0811.60098
  8. DERRIDA, B., Random energy model: An exactly solvable model of disordered systems. Phys. Rev., B 24 # 5, 1981, 2613-2626. Zbl1323.60134
  9. DERRIDA, B. - GARDNER, E., Optimal storage properties of neural network models. J. Phys., A 21, 1988, 271-284. 
  10. FRÖHLICH, J. - ZEGARLINSKI, B., Some comments on the Sherrington-Kirkpatrick model of spin glasses. Commun. Math. Phys., 112, 1987, 553-566. 
  11. GARDNER, E., Spin glasses with p-spin interactions. Nuclear Phys., B 257, #6, 1985, 747-765. 
  12. GARDNER, E., The space of interactions in neural network models. J. Phys., A 21, 1988, 257-270. Zbl1128.82302
  13. HOPFIELD, J. J., Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA, 79, 1982, 1554-2558. Zbl1369.92007
  14. KRAUTH, W. - MÉZARD, M., Storage capacity of memory network with binary couplings. J. Phys., 50, 1989, 3057-3066. 
  15. MÉZARD, M., The space of interactions in neural networks: Gardner's computation with the cavity method. J. Phys., A 22, 1988, 2181-2190. 
  16. MÉZARD, M. - PARISI, G. - VIRASIRO, M., Spin Glass Theory and Beyond. World Scientific, Singapore1987. 
  17. MILMAN, V., The heritage of P. Levy in functional analysis. Colloque P. Lévy sur les processus stochastiques, Astérisque, 157-158, 1988, 273-301. 
  18. MILMAN, V. - SCHECHTMAN, G., Asymptotic Theory of Finite Dimensional Normed Spaces. Lecture Notes in Math., 1200, Springer Verlag, 1986. Zbl0911.52002
  19. PARISI, G., Field Theory, Disorder, Simulation. World Scientific Lecture Notes in Physics, 45, World Scientific, Singapore1992. 
  20. SCHCHERBINA, M., On the replica-symmetric solution for the Sherrington-Kirkpatrick Model. Helv. Phys. Acta., 70, 1997, 838-853. Zbl0899.60096
  21. SHERRINGTON, D. - KIRKPATRICK, S., Solvable model of a spin glass. Phys. Rev. Lett., 35, 1972, 1792-1796. 
  22. TALAGRAND, M., Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. I.H.E.S., 81, 1995, 73-205. Zbl0864.60013
  23. TALAGRAND, M., A new look at independence. Ann. Probab., 24, 1996, 1-34. Zbl0858.60019
  24. TALAGRAND, M., The Sherrington-Kirkpatrick model: a challenge to mathematicians. Probab. Th. Rel. Fields, 110, 1998, 109-176. Zbl0909.60083
  25. TALAGRAND, M., Rigorous results for the Hopfield model with many patterns. Probab. Th. Rel. Fields, 110, 1998, 177-276. Zbl0897.60041
  26. TALAGRAND, M., Rigorous low temperature results for the p-spin interaction model. Probab. Th. Rel. Fields, 117, 2000, 303-360. Zbl1016.82021
  27. TALAGRAND, M., The high temperature case of the K-sat problem. Probab. Th. Rel. Fields, to appear Zbl0999.82045
  28. TALAGRAND, M., Self averaging and the space of interactions in neural networks. Random Structures and Algorithms, 14, 1998, 199-213. Zbl0921.60098
  29. TALAGRAND, M., Exponential inequalities and replica-symmetry breaking for the Sherrington-Kirkpatrick model. Ann. Probab., to appear. Zbl1034.82027MR1797303DOI10.1214/aop/1019160325
  30. TALAGRAND, M., Exponential inequalities and convergence of moments in the replica-symmetric regime of the Hopfield model. Ann. Probab., to appear. Zbl1043.82036MR1813831DOI10.1214/aop/1019160495
  31. TALAGRAND, M., Intersecting random half spaces: towards the Gardner-Derrida formula. Ann. Probab., to appear. Zbl1043.82030MR1782273DOI10.1214/aop/1019160259
  32. TALAGRAND, M., Intersecting random half cubes. Random Structures and Algorithms, 15, 1999, 436-449. Zbl0936.60007MR1716771DOI10.1002/(SICI)1098-2418(199910/12)15:3/4<436::AID-RSA11>3.0.CO;2-5
  33. TALAGRAND, M., Huge random structures and mean field models for spin glasses. Proceedings of the International Congress of mathematicians, Documenta Math. Extra, vol. I, 1998. Zbl0902.60089MR1648045
  34. TALAGRAND, M., Verres de spin et optimisation combinatoire. Séminaire Bourbaki, Vol. 1998/99, Exposés 850-864, Astérisque, 266, 2000, 287-317. Zbl1005.82014MR1772677

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