The Annals of Applied Probability

The K-process on a tree as a scaling limit of the GREM-like trap model

L. R. G. Fontes, R. J. Gava, and V. Gayrard

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Abstract

We introduce trap models on a finite volume $k$-level tree as a class of Markov jump processes with state space the leaves of that tree. They serve to describe the GREM-like trap model of Sasaki and Nemoto. Under suitable conditions on the parameters of the trap model, we establish its infinite volume limit, given by what we call a $K$-process in an infinite $k$-level tree. From this we deduce that the $K$-process also is the scaling limit of the GREM-like trap model on extreme time scales under a fine tuning assumption on the volumes.

Article information

Source
Ann. Appl. Probab., Volume 24, Number 2 (2014), 857-897.

Dates
First available in Project Euclid: 10 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1394465373

Digital Object Identifier
doi:10.1214/13-AAP937

Mathematical Reviews number (MathSciNet)
MR3178499

Zentralblatt MATH identifier
1302.60133

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C44: Dynamics of disordered systems (random Ising systems, etc.)

Keywords
Random dynamics random environments $K$-process scaling limit trap models GREM

Citation

Fontes, L. R. G.; Gava, R. J.; Gayrard, V. The K -process on a tree as a scaling limit of the GREM-like trap model. Ann. Appl. Probab. 24 (2014), no. 2, 857--897. doi:10.1214/13-AAP937. https://projecteuclid.org/euclid.aoap/1394465373


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References

  • [1] Barlow, M. T. and Černý, J. (2011). Convergence to fractional kinetics for random walks associated with unbounded conductances. Probab. Theory Related Fields 149 639–673.
  • [2] 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.
  • [3] 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.
  • [4] 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.
  • [5] Ben Arous, G. and Gün, O. (2012). Universality and extremal aging for dynamics of spin glasses on subexponential time scales. Comm. Pure Appl. Math. 65 77–127.
  • [6] Ben Arous, G. and Černý, J. (2007). Scaling limit for trap models on $\mathbb{Z}^{d}$. Ann. Probab. 35 2356–2384.
  • [7] 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.
  • [8] Ben Arous, G., Černý, J. and Mountford, T. (2006). Aging in two-dimensional Bouchaud’s model. Probab. Theory Related Fields 134 1–43.
  • [9] Bezerra, S. C., Fontes, L. R. G., Gava, R. J., Gayrard, V. and Mathieu, P. (2012). Scaling limits and aging for asymmetric trap models on the complete graph and $K$ processes. ALEA Lat. Am. J. Probab. Math. Stat. 9 303–321.
  • [10] Bouchaud, J. P. and Dean, D. S. (1995). Aging on Parisi’s tree. J. Phys. I France 5 265–286.
  • [11] Bovier, A. and Faggionato, A. (2005). Spectral characterisation of ageing: The REM-like trap model in the complete graph. Ann. Appl. Probab. 15 1997–2037.
  • [12] Bovier, A. and Gayrard, V. (2013). Convergence of clock processes in random environments and ageing in the $p$-spin SK model. Ann. Probab. 41 817–847.
  • [13] Bovier, A., Gayrard, V. and Švejda, A. (2013). Convergence to extremal processes in random environments and extremal ageing in SK models. Probab. Theory Related Fields 157 251–283.
  • [14] Compte, A. and Bouchaud, J. P. (1998). Localization in one-dimensional random walks. J. Phys. A: Math. Gen. 31 6113–6121.
  • [15] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [16] Fontes, L. R. G., Isopi, M. and Newman, C. M. (2002). Random walks with strongly inhomogeneous rates and singular diffusions: Convergence, localization and aging in one dimension. Ann. Probab. 30 579–604.
  • [17] Fontes, L. R. G. and Lima, P. H. S. (2009). Convergence of symmetric trap models in the hypercube. In XVth International Congress on Mathematical Physics, 2006, Rio de Janeiro. New Trends in Mathematical Physics 285–297. Springer, Dordrecht.
  • [18] Fontes, L. R. G. and Mathieu, P. (2008). $K$-processes, scaling limit and aging for the trap model in the complete graph. Ann. Probab. 36 1322–1358.
  • [19] Gayrard, V. (2010). Aging in reversible dynamics of disordered systems. I. Emergence of the arcsine law in Bouchaud’s asymmetric trap model on the complete graph. Available at arXiv:1008.3855v1 [math.PR] (longer version of [21]).
  • [20] 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. Available at arXiv:1008.3849.
  • [21] 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.
  • [22] Gayrard, V. and Gün, O. (2013). In preparation.
  • [23] LePage, R., Woodroofe, M. and Zinn, J. (1981). Convergence to a stable distribution via order statistics. Ann. Probab. 9 624–632.
  • [24] Nieuwenhuizen, T. M. and Ernst, M. H. (1985). Excess noise in a hopping model for a resistor with quenched disorder. J. Stat. Phys. 41 773–801.
  • [25] Sasaki, M. and Nemoto, K. (2000). Analysis on aging in the generalized random energy model. J. Phys. Soc. Jpn. 69 3045–3050.
  • [26] Sasaki, M. and Nemoto, K. (2001). Numerical study of aging in the generalized random energy model. J. Phys. Soc. Jpn. 70 1099–1104.