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Nature Versus Nurture: Dynamical Evolution in Disordered Ising Ferromagnets

  • Lily Z. Wang
  • Reza GheissariEmail author
  • Charles M. Newman
  • Daniel L. Stein
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 293)

Abstract

We study the predictability of zero-temperature Glauber dynamics in various models of disordered ferromagnets. This is analyzed using two independent dynamical realizations with the same random initialization (called twins). We derive, theoretically and numerically, trajectories for the evolution of the normalized magnetization and twin overlap as the system size tends to infinity. The systems we treat include mean-field ferromagnets with light-tailed and heavy-tailed coupling distributions, as well as highly-disordered models with a variety of other geometries. In the mean-field setting with light-tailed couplings, the disorder averages out and the limiting trajectories of the magnetization and twin overlap match those of the homogenous Curie–Weiss model. On the other hand, when the coupling distribution has heavy tails, or the geometry changes, the effect of the disorder persists in the thermodynamic limit. Nonetheless, qualitatively all such random ferromagnets share a similar time evolution for their twin overlap, wherein the two twins initially decorrelate, before either partially or fully converging back together due to the ferromagnetic drift.

Keywords

Glauber dynamics Predictability Random ferromagnet Curie–Weiss model Spin glass 

Notes

Acknowledgements

We thank the anonymous referee for helpful suggestions. The research of CMN was supported in part by U.S. NSF Grant DMS-1507019. DLS thanks the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.

References

  1. 1.
    Ben Arous, G., Gheissari, R., Jagannath, A.: Algorithmic thresholds for tensor PCA (2018). arXiv:1808.00921 (preprint)
  2. 2.
    Ben Arous, G., Mei, S., Montanari, A., Nica, M.: The landscape of the spiked tensor model (2017). arXiv:1711.05424
  3. 3.
    Bray, A.J.: Theory of phase-ordering kinetics. Adv. Phys. 43, 357–459 (1994)CrossRefGoogle Scholar
  4. 4.
    Crisanti, A., Leuzzi, L.: Spherical \(2+p\) spin-glass model: an exactly solvable model for glass to spin-glass transition. Phys. Rev. Lett. 93, 217203 (2004)CrossRefGoogle Scholar
  5. 5.
    Crisanti, A., Leuzzi, L.: Spherical \(2+p\) spin-glass model: an analytically solvable model with a glass-to-glass transition. Phys. Rev. B 73, 014412 (2006)CrossRefGoogle Scholar
  6. 6.
    Gheissari, R., Newman, C.M., Stein, D.L.: Zero-temperature dynamics in the dilute Curie–Weiss model. J. Stat. Phys. 172(4), 1009–1028 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gillin, P., Sherrington, D.: \(p > 2\) spin glasses with first-order ferromagnetic transitions. J. Phys. A: Math. Gen. 33(16), 3081 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Haggstrom, O.: Zero-temperature dynamics for the ferromagnetic Ising model on random graphs. Phys. A: Stat. Mech. Appl. 310(3), 275–284 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Nanda, S., Newman, C.M.: Random nearest neighbor and influence graphs on \({ {z}}^d\). Rand. Struct. Alg. 15, 262–278 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Nanda, S., Newman, C.M., Stein, D.L.: Dynamics of Ising spin systems at zero temperature. In: Minlos, R., Shlosman, S., Suhov, Y. (eds.) On Dobrushin’s Way (from Probability Theory to Statistical Physics). Amer. Math. Soc. Transl. 198(2), pp. 183–194 (2000)Google Scholar
  11. 11.
    Newman, C.M., Stein, D.L.: Spin-glass model with dimension-dependent ground state multiplicity. Phys. Rev. Lett. 72, 2286–2289 (1994)CrossRefGoogle Scholar
  12. 12.
    Newman, C.M., Stein, D.L.: Ground state structure in a highly disordered spin glass model. J. Stat. Phys. 82, 1113–1132 (1996)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Richard, E., Montanari, A.: A statistical model for tensor PCA. In: Advances in Neural Information Processing Systems, pp. 2897–2905 (2014)Google Scholar
  14. 14.
    Ros, V., Ben Arous, G., Biroli, G., Cammarota, C.: Complex energy landscapes in spiked-tensor and simple glassy models: ruggedness, arrangements of local minima and phase transitions (2018). arXiv:1804.02686
  15. 15.
    Song, Y., Gheissari, R., Newman, C.M., Stein, D.L.: Searching for local minima in a random landscape (2019, in preparation)Google Scholar
  16. 16.
    Stein, D.L., Newman, C.M.: Nature versus nurture in complex and not-so-complex systems. In: Sanayei, A., Zelinka, I., Rössler, O. (eds.) ISCS 2013: Interdisciplinary Symposium on Complex Systems, pp. 57–63. Springer, Berlin (2014)CrossRefGoogle Scholar
  17. 17.
    Ye, J., Gheissari, R., Machta, J., Newman, C.M., Stein, D.L.: Long-time predictability in disordered spin systems following a deep quench. Phys. Rev. E 95, 042101 (2017)CrossRefGoogle Scholar
  18. 18.
    Ye, J., Machta, J., Newman, C.M., Stein, D.L.: Nature versus nurture: predictability in low-temperature Ising dynamics. Phys. Rev. E 88, 040101 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Lily Z. Wang
    • 1
  • Reza Gheissari
    • 2
    Email author
  • Charles M. Newman
    • 2
    • 3
  • Daniel L. Stein
    • 4
    • 5
    • 6
  1. 1.Center for Applied Mathematics, Cornell UniversityIthacaUSA
  2. 2.Courant Institute, New York UniversityNew YorkUSA
  3. 3.NYU-ECNU Institute of Mathematical Sciences at NYU ShanghaiShanghaiChina
  4. 4.Department of Physics and Courant InstituteNew York UniversityNew YorkUSA
  5. 5.NYU-ECNU Institutes of Physics and Mathematical Sciences at NYU ShanghaiShanghaiChina
  6. 6.Santa Fe InstituteSanta FeUSA

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