Abstract
We study the Langevin dynamics for the family of spherical p-spin disordered mean-field models of statistical physics. We prove that in the limit of system size N approaching infinity, the empirical state correlation and integrated response functions for these N-dimensional coupled diffusions converge almost surely and uniformly in time, to the non-random unique strong solution of a pair of explicit non-linear integro-differential equations intensively studied by Cugliandolo and Kurchan.
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Aida, S., Stroock, D.: Moment estimates derived from Poincaré and logarithmic Sobolev inequalities. Math. Res. Lett. 1, 75–86 (1994)
Ané, C. et altri: Sur les inégalités de Sobolev logarithmiques. Panoramas et Syntheses, 10, Société Mathématique de France (2000).
Ben Arous, G.: Aging and spin-glass dynamics. Proceedings of the International Congress of Mathematicians, Vol. III , 3–14, Higher Ed. Press, Bei**g, 2002
Ben Arous, G., Bovier, A., Gayrard, V.: Glauber dynamics of the random energy model. II. Aging below the critical temperature. Comm. Math. Phys. 236, 1–54 (2003)
Ben Arous, G., Dembo, A., Guionnet, A.: Aging of spherical spin glasses, Prob. Th. Rel. Fields. 120, 1–67 (2001)
Ben Arous, G., Guionnet, A.: Large deviations for Langevin spin glass dynamics. Prob. Th. Rel. Fields 102, 455–509 (1995)
Ben Arous, G., Guionnet, A.: Symmetric Langevin spin glass dynamics. Ann. Probab. 25, 1367–1422 (1997)
Bouchaud, J.P., Cugliandolo, L.F., Kurchan, J., Mezard, M.: Out of equilibrium dynamics in spin-glasses and other glassy systems. In: Young, A.P. (ed.) Spin glass dynamics and Random Fields 1997
Bovier, A., Gayrard, V.: The retrieval phase of the Hopfield model: a rigorous analysis of the overlap distribution. Probab. Theory Related Fields. 107, 61–98 (1997)
Crisanti, H., Horner, H., Sommers, H.-J.: The Spherical p-spin Interaction Spin Glass Model: The Dynamics, Zeitschrift fur Physik B 92, 257–271 (1993)
Cugliandolo, L.F.: Dynamics of glassy systems. In: Barrat, J-L. et al. (eds.) Slow relaxations and non-equilibirum dynamics in condensed matter. Springer, Berlin, 2003
Cugliandolo, L.F., Dean, D.S.: Full dynamical solution for a spherical spin-glass model. J. Phys. A: Math. Gen. 28, 4213–4234 (1995)
Cugliandolo, L.F., Kurchan, J.: Analytical solution of the off-equilibrium Dynamics of a Long-Range Spin-Glass Model. Phys. Rev. Lett. 71, 173 (1993)
Dudley, R.: The sizes of compact subsets of Hilbert space and continuity of Gaussian Processes. J. Funct. Analysis 1, 290–330 (1967)
Grunwald, M.: Sanov results for Glauber spin-glass dynamics. Prob. Th. Rel. Fields 106, 187–232 (1996)
Guionnet, A.: Annealed and quenched propagation of chaos for Langevin spin glass dynamics. Prob. Th. Rel. Fields. 109, 183–215 (1997)
Guionnet, A., Mazza, C.: Long time behaviour of non-commutative processes solution of a linear differential equation. Prob. Theory. Rel. Fields. 131, 493–518 (2005)
Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus, 2nd edn. Springer-Verlag, 1991
Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Ergebnisse der Mathematik 23, Springer Verlag, 1991
Mezard, M., Parisi, G.: Virasoro, M. ; Spin glass theory and beyond. World Scientific Lecture Notes in Physic, 1987
Sompolinsky, H., Zippelius, A.: Relaxational dynamics of the Edwards-Anderson model and the mean-field theory of spin glasses. Phys. Rev. B. 25, 6860–6875 (1982)
Talagrand, M.: Spin Glasses: a Challenge for Mathematicians. Ergebnisse der Mathematik 46, Springer Verlag, 2003
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Research partially supported by NSF grants #DMS-0406042, #DMS-FRG-0244323
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Ben Arous, G., Dembo, A. & Guionnet, A. Cugliandolo-Kurchan equations for dynamics of Spin-Glasses. Probab. Theory Relat. Fields 136, 619–660 (2006). https://doi.org/10.1007/s00440-005-0491-y
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DOI: https://doi.org/10.1007/s00440-005-0491-y