Electronic Journal of Probability

Dynamical freezing in a spin glass system with logarithmic correlations

Aser Cortines, Julian Gold, and Oren Louidor

Full-text: Open access

Abstract

We consider a continuous time random walk on the two-dimensional discrete torus, whose motion is governed by the discrete Gaussian free field on the corresponding box acting as a potential. More precisely, at any vertex the walk waits an exponentially distributed time with mean given by the exponential of the field and then jumps to one of its neighbors, chosen uniformly at random. We prove that throughout the low-temperature regime and at in-equilibrium timescales, the process admits a scaling limit as a spatial K-process driven by a random trapping landscape, which is explicitly related to the limiting extremal process of the field. Alternatively, the limiting process is a supercritical Liouville Brownian motion with respect to the continuum Gaussian free field on the box. This demonstrates rigorously and for the first time, as far as we know, a dynamical freezing in a spin glass system with logarithmically correlated energy levels.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 59, 31 pp.

Dates
Received: 26 November 2017
Accepted: 23 May 2018
First available in Project Euclid: 11 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1528704077

Digital Object Identifier
doi:10.1214/18-EJP181

Mathematical Reviews number (MathSciNet)
MR3814253

Zentralblatt MATH identifier
06924671

Subjects
Primary: 60K37: Processes in random environments 82C44: Dynamics of disordered systems (random Ising systems, etc.) 60G57: Random measures 60G70: Extreme value theory; extremal processes 60G15: Gaussian processes 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Keywords
K-process trap models aging Gaussian free field random walk in a random potential dynamical freezing spin-glasses

Rights
Creative Commons Attribution 4.0 International License.

Citation

Cortines, Aser; Gold, Julian; Louidor, Oren. Dynamical freezing in a spin glass system with logarithmic correlations. Electron. J. Probab. 23 (2018), paper no. 59, 31 pp. doi:10.1214/18-EJP181. https://projecteuclid.org/euclid.ejp/1528704077


Export citation

References

  • [1] P. Auer, The circle homogeneously covered by random walk on $\mathbb{Z} ^2$, Statist. Probab. Lett. 9 (1990), no. 5, 403–407.
  • [2] J. Barral, X. Jin, R. Rhodes, and V. Vargas, Gaussian multiplicative chaos and KPZ duality, Comm. Math. Phys. 323 (2013), no. 2, 451–485.
  • [3] J. Beltran and C. Landim, Tunneling and metastability of continuous time Markov chains, J. Stat. Phys. 140 (2010), no. 6, 1065–1114.
  • [4] G. Ben Arous, A. Bovier, and V. Gayrard, Glauber dynamics of the random energy model. I. Metastable motion on the extreme states, Comm. Math. Phys. 235 (2003), no. 3, 379–425.
  • [5] G. Ben Arous, A. Bovier, and V. Gayrard, Glauber dynamics of the random energy model. II. Aging below the critical temperature, Comm. Math. Phys. 236 (2003), no. 1, 1–54.
  • [6] G. Ben Arous and J. Černý, Scaling limit for trap models on $\mathbb{Z} ^d$, Ann. Probab. 35 (2007), no. 6, 2356–2384.
  • [7] G. Ben Arous and J. Černý, The arcsine law as a universal aging scheme for trap models, Comm. Pure Appl. Math. 61 (2008), no. 3, 289–329.
  • [8] N. Berestycki, Diffusion in planar Liouville quantum gravity, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), no. 3, 947–964.
  • [9] M. Biskup, Recent progress on the random conductance model, Probab. Surv. 8 (2011), 294–373.
  • [10] M. Biskup, J. Ding, and S. Goswami, Return probability and recurrence for the random walk driven by two-dimensional Gaussian free field, arXiv:1611.03901 (2016).
  • [11] M. Biskup and O. Louidor, Full extremal process, cluster law and freezing for two-dimensional discrete Gaussian free field, arXiv:1606.00510 (2016).
  • [12] M. Biskup and O. Louidor, On intermediate level sets of two-dimensional discrete Gaussian free field, arXiv:1612.01424 (2016).
  • [13] A. Bovier and V. Gayrard, Convergence of clock processes in random environments and ageing in the $p$-spin SK model, Ann. Probab. 41 (2013), no. 2, 817–847.
  • [14] M. Bramson, J. Ding, and O. Zeitouni, Convergence in law of the maximum of the two-dimensional discrete Gaussian free field, Comm. Pure Appl. Math. 69 (2016), no. 1, 62–123.
  • [15] D. Carpentier and P. Le Doussal, Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in Liouville and sinh-Gordon models, Phys. Rev. E 63 (2001), no. 2, 026110.
  • [16] H. E. Castillo and P. Le Doussal, Freezing of dynamical exponents in low dimensional random media, Phys. Rev. Lett. 86 (2001), no. 21, 4859.
  • [17] O. Daviaud, Extremes of the discrete two-dimensional Gaussian free field, Ann. Probab. 34 (2006), no. 3, 962–986.
  • [18] J. Ding and O. Zeitouni, Extreme values for two-dimensional discrete Gaussian free field, Ann. Probab. 42 (2014), no. 4, 1480–1515.
  • [19] L. R. Fontes, R. J. Gava, and V. Gayrard, 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.
  • [20] L. R. Fontes and P. Mathieu, K-processes, scaling limit and aging for the trap model in the complete graph, Ann. Probab. 36 (2008), no. 4, 1322–1358.
  • [21] L. R. Fontes and G. R. C. Peixoto, GREM-like K processes on trees with infinite depth, arXiv:1412.4291 (2014).
  • [22] C. Garban, R. Rhodes, and V. Vargas, Liouville Brownian motion, Ann. Probab. 44 (2016), no. 4, 3076–3110.
  • [23] M. Jara, C. Landim, and A. Teixeira, Quenched scaling limits of trap models, Ann. Probab. 39 (2011), no. 1, 176–223.
  • [24] M. Jara, C. Landim, and A. Teixeira, Universality of trap models in the ergodic time scale, Ann. Probab 42 (2014), no. 6, 2497–2557.
  • [25] A. N. Kolmogorov, On the differentiability of the transition probabilities in stationary Markov processes with a denumberable number of states, Uchenye Zapiski Moskovskogo Gosudarstvennogo Universiteta 148 (1951), 53–59.
  • [26] G. F. Lawler and V. Limic, Random walk: a modern introduction, Cambridge Studies in Advanced Mathematics, vol. 123, Cambridge University Press, Cambridge, 2010.
  • [27] B. Morris and Y. Peres, Evolving sets, mixing and heat kernel bounds, Probab. Theory Related Fields 133 (2005), no. 2, 245–266.
  • [28] R. Rhodes and V. Vargas, Gaussian multiplicative chaos and applications: a review, Probab. Surv. 11 (2014), 315–392.
  • [29] R. Rhodes and V. Vargas, Liouville Brownian motion at criticality, Potential Anal. 43 (2015), no. 2, 149–197.
  • [30] Ya. G Sinai, The limiting behavior of a one-dimensional random walk in a random medium, Theory Probab. Appl. 27 (1983), no. 2, 256–268.