## Convergence of clock processes in random environments and ageing in the $$p$$-spin SK model.(English)Zbl 1267.82114

Summary: We derive a general criterion for the convergence of clock processes in random dynamics in random environments which is applicable in cases in which correlations are not negligible, extending recent results [the second author, Electron. J. Probab. 17, Paper No. 58 (2012; Zbl 1252.82091); “Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM”, Preprint, arXiv:1008.3849], based on general criterion for convergence of sums of dependent random variables due to R. Durrett and S. I. Resnick [Ann. Probab. 6, 829–846 (1978; Zbl 0398.60024)]. We demonstrate the power of this criterion by applying it to the random hopping time dynamics of the $$p$$-spin SK model. We prove that, for a wide range of time scales, the clock process converges to a stable subordinator almost surely with respect to the environment. We also show that a time-time correlation function converges to the arcsine law for this subordinator, almost surely. This improves recent similar convergence results in law with respect to the random environment (see [G. Ben Arousand and the authors, Commun. Math. Phys. 236, No. 1, 1–54 (2003; Zbl 1037.82039)]).

### MSC:

 82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G70 Extreme value theory; extremal stochastic processes

### Citations:

Zbl 1252.82091; Zbl 0398.60024; Zbl 1037.82039
Full Text:

### References:

 [1] Ben Arous, G., Bovier, A. and Černý, J. (2008). Universality of the REM for dynamics of mean-field spin glasses. Comm. Math. Phys. 282 663-695. · Zbl 1208.82024 [2] Ben Arous, G., Bovier, A. and Gayrard, V. (2002). Aging in the random energy model. Phys. Rev. Lett. 88 087201. · Zbl 1037.82039 [3] Ben Arous, G., Bovier, A. and Gayrard, V. (2003). Glauber dynamics of the random energy model. I. Metastable motion on the extreme states. Comm. Math. Phys. 235 379-425. · Zbl 1037.82038 [4] Ben Arous, G., Bovier, A. and Gayrard, V. (2003). Glauber dynamics of the random energy model. II. Aging below the critical temperature. Comm. Math. Phys. 236 1-54. · Zbl 1037.82039 [5] Ben Arous, G. and Černý, J. (2005). Bouchaud’s model exhibits two different aging regimes in dimension one. Ann. Appl. Probab. 15 1161-1192. · Zbl 1069.60092 [6] Ben Arous, G. and Černý, J. (2006). Dynamics of trap models. In Mathematical Statistical Physics 331-394. Elsevier, Amsterdam. · Zbl 1458.82019 [7] Ben Arous, G. and Černý, J. (2007). Scaling limit for trap models on $$\mathbb{Z}^{d}$$. Ann. Probab. 35 2356-2384. · Zbl 1134.60064 [8] Ben Arous, G. and Černý, J. (2008). The arcsine law as a universal aging scheme for trap models. Comm. Pure Appl. Math. 61 289-329. · Zbl 1141.60075 [9] Ben Arous, G., Černý, J. and Mountford, T. (2006). Aging in two-dimensional Bouchaud’s model. Probab. Theory Related Fields 134 1-43. · Zbl 1089.82017 [10] Bouchaud, J. P. (1992). Weak ergodicity breaking and aging in disordered systems. J. Phys. I ( France ) 2 1705-1713. [11] Bouchaud, J. P., Cugliandolo, L., Kurchan, J. and Mézard, M. (1998). Out of equilibrium dynamics in spin-glasses and other glassy systems. In Spin Glasses and Random Fields (A. P. Young, ed.). World Scientific, Singapore. [12] Bouchaud, J. P. and Dean, D. S. (1995). Aging on Parisi’s tree. J. Phys. I ( France ) 5 265. [13] Durrett, R. and Resnick, S. I. (1978). Functional limit theorems for dependent variables. Ann. Probab. 6 829-846. · Zbl 0398.60024 [14] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II , 2nd ed. Wiley, New York. · Zbl 0219.60003 [15] Gayrard, V. (2012). Convergence of clock process in random environments and aging in Bouchaud’s asymmetric trap model on the complete graph. Electron. J. Probab. 17 1-33. · Zbl 1252.82091 [16] Gayrard, V. (2010). Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM. Preprint. Available at . 1008.3849 [17] Gayrard, V. (2011). Aging in reversible dynamics of disordered systems. III. Emergence of the arcsine law in the Metropolis dynamics of the REM. Preprint in preparation, LAPT, Marseille. · Zbl 1252.82091 [18] Gnedenko, B. V. and Kolmogorov, A. N. (1949). Predel’nye Raspredeleniya Dlya Summ Nezavisimyh Slučaĭ nyh Veličin . Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad. [19] Goldstein, M. (1969). Viscous liquids and the glass transition: A potential energy barrier picture. The Journal of Chemical Physics 51 3728-3739. [20] Kemperman, J. H. B. (1974). The Passage Problem for a Stationary Markov Chain . Reidel, Dordrecht, Holland. · Zbl 0326.60081 [21] Monthus, C. and Bouchaud, J. P. (1996). Models of traps and glass phenomenology. J. Phys. A 29 3847-3869. · Zbl 0900.82090 [22] Rinn, B., Maass, P. and Bouchaud, J. P. (2000). Multiple scaling regimes in simple aging models. Phys. Rev. Lett. 84 5403-5406. [23] Rogers, L. C. G. and Williams, D. (2000). Diffusions , Markov Processes , and Martingales. Vol. 1. Cambridge Mathematical Library . Cambridge Univ. Press, Cambridge. · Zbl 0977.60005 [24] Sinaĭ, Y. G. (1982). The limit behavior of a one-dimensional random walk in a random environment. Teor. Veroyatn. Primen. 27 247-258. [25] Solomon, F. (1975). Random walks in a random environment. Ann. Probab. 3 1-31. · Zbl 0305.60029 [26] Whitt, W. (2002). Stochastic-Process Limits : An Introduction to Stochastic-Process Limits and Their Application to Queues . Springer, New York. · Zbl 0993.60001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.