Abstract
Let E x be a collection of i.i.d. exponential random variables. Symmetric Bouchaud's model on ℤ2 is a Markov chain X(t) whose transition rates are given by w xy = ν exp (−βE x ) if x, y are neighbours in ℤ2. We study the behaviour of two correlation functions: ℙ[X(t w +t) = X(t w )] and ℙ[X(t') = X(t w ) ∀ t'∈ [t w , t w + t]]. We prove the (sub)aging behaviour of these functions when β > 1.
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Ben Arous, G., Bovier, A., Gayrard, V.: Glauber dynamics of the random energy model. I. Metastable motion on the extreme states. Comm. Math. Phys. 235 (3), 379–425 (2003)
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), 1–54 (2003)
Ben Arous, G., Černý, J.: Bouchaud's model exhibits two aging regimes in dimension one. To appear in Annals of Applied Probability (2004)
Ben Arous, G.: Aging and spin glass dynamics. Proceedings of Inter. Cong. Matematicians. Beijing 2002 III, 1–12 (2002)
Bertoin, J.: Lévy processes. Cambridge: Cambridge University Press, 1996
Billingsley, P. Convergence of probability measures. second ed., John Wiley & Sons Inc., New York, 1999, A Wiley-Interscience Publication
Bouchaud, J.-P., Mézard, M.: Universality classes for extreme-value statistics. J. Phys. A: Math. Gen. 30, 7997–8015 (1997)
Bouchaud, J.-P.: Weak ergodicity breaking and aging in disordered systems. J. Phys. I (France) 2, 1705–1713 (1992)
Černý, J.: On two properties of strongly disordered systems, aging and critical path analysis. Ph.D. thesis, EPF Lausanne, 2003
Fontes, L.R.G., Isopi, M., Newman, C.M.: Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension. Ann. Probab. 30 (2), 579–604 (2002)
Lawler, G.F.: Intersections of random walks. Birkhäuser Boston Inc., Boston, MA, 1991
Monthus, C., Bouchaud, J.-P.: Models of traps and glass phenomenology. J. Phys. A 29, 3847–3869 (1996)
Rinn, B., Maass, P., Bouchaud, J.-P.: Multiple scaling regimes in simple aging models. Phys. Rev. Lett 84, 5403–5406 (2000)
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Ben Arous, G., Černý, J. & Mountford, T. Aging in two-dimensional Bouchaud's model. Probab. Theory Relat. Fields 134, 1–43 (2006). https://doi.org/10.1007/s00440-004-0408-1
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DOI: https://doi.org/10.1007/s00440-004-0408-1