Non-fixation for Conservative Stochastic Dynamics on the Line

Abstract

We consider activated random walk (ARW), a model which generalizes the stochastic sandpile, one of the canonical examples of self organized criticality. Informally ARW is a particle system on \({\mathbb{Z}}\) with mass conservation. One starts with a mass density \({\mu > 0}\) of initially active particles, each of which performs a symmetric random walk at rate one and falls asleep at rate \({\lambda > 0}\). Sleepy particles become active on coming in contact with other active particles. We investigate the question of fixation/non-fixation of the process and show for small enough \({\lambda}\) the critical mass density for fixation is strictly less than one. Moreover, the critical density goes to zero as \({\lambda}\) tends to zero. This settles a long standing open question.

This is a preview of subscription content, log in to check access.

References

  1. 1

    Amir, G., Gurel-Gurevich, O.: On fixation of activated random walks. Electron. Commun. Probab. 12, 119–123 (2010)

  2. 2

    Andjel E.D.: Invariant measures for the zero range processes. Ann. Probab. 10, 525–547 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3

    Bak P., Tang C., Wiesenfeld K.: Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. Lett. 59, 381–384 (1987)

    ADS  Article  Google Scholar 

  4. 4

    Basu M., Basu U., Bandyopadhyay S., Mohanty P.K., Hinrichsen H.: Fixed-energy sandpiles belong generically to directed percolation. Phys. Rev. Lett. 109, 44–48 (2012)

    Article  Google Scholar 

  5. 5

    Bonachela J.A., Muñoz M.A.: Confirming and extending the hypothesis of universality in sandpiles. Phys. Rev. E 78, 041102 (2008)

    ADS  Article  Google Scholar 

  6. 6

    Bond, B., Levine, L.: Abelian Networks I. Foundations and Examples. arXiv:1309.3445 (2013)

  7. 7

    Cabezas, M., Rolla, L.T., Sidoravicius, V.: Recurrence and density decay for diffusion limited annihilating systems. Probab. Theor. Relat. Fields. arXiv:1309.4387 (to appear)

  8. 8

    Cabezas M., Rolla L.T., Sidoravicius V.: Non-equilibrium phase transitions: activated random walks at criticality. J. Stat. Phys. 155(6), 1112–1125 (2014)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  9. 9

    da Cunha D.S., Vidigal R.R., da Silva R.L., Dickman R.: Diffusion in stochastic sandpiles. Eur. Phys. J. B 72(3), 441–449 (2009)

    ADS  Article  MATH  Google Scholar 

  10. 10

    Dhar D.: The abelian sandpile and related models. Phys. A 263, 4–25 (1999)

    Article  Google Scholar 

  11. 11

    Diaconis P., Fulton W.: A growth model, a game, an algebra, Lagrange inversion, and characteristic classes. Rend. Sem. Mat. Univ. Politec. Torino 49(1), 95–119 (1991)

    MathSciNet  MATH  Google Scholar 

  12. 12

    Dickman R.: Nonequilibrium phase transitions in epidemics and sandpiles. Phys. A 306, 90–97 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13

    Dickman R., Alava M., Muñoz M.A., Peltola J., Vespignani A., Zapperi S.: Critical behaviour of a one-dimensional fixed-energy sandpile. Phys. Rev. E 64, 56104 (2001)

    ADS  Article  Google Scholar 

  14. 14

    Dickman R., Muñoz M.A., Vespignani A., Zapperi S.: Paths to self-organized criticality. Braz. J. Phys. 30, 27 (2000)

    ADS  Article  Google Scholar 

  15. 15

    Dickman R., Vespignani A., Zapperi S.: Self-organized criticality as an absorbing-state phase transition. Phys. Rev. E 57, 5095–5105 (1998)

    ADS  Article  Google Scholar 

  16. 16

    Dickman R., Rolla L.T., Sidoravicius V.: Activated random walkers: facts, conjectures and challenges. J. Stat. Phys. 138(1–3), 126–142 (2010)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. 17

    Eriksson K.: Chip-firing games on mutating graphs. SIAM J. Discret. Math. 9(1), 118–128 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18

    Fey A., Levine L., Wilson D.B.: Driving sandpiles to criticality and beyond. Phys. Rev. Lett. 104(14), 145703 (2010)

    ADS  Article  Google Scholar 

  19. 19

    Holroyd, A.E., Levine, L., Mészáros, K., Peres, Y., Propp, J., Wilson, D.B.: Chip-firing and rotor-routing on directed graphs. In: Sidoravicius, V., Vares, M.E. (eds) In and Out of Equilibrium 2, pp. 331–364. Birkhäuser, Basel, Switzerland (2008)

  20. 20

    Hough, B., Jerison, D., Levine, L.: Sandpiles on the Square Lattice. arXiv:1703.00827 (2017)

  21. 21

    Jain K.: Simple sandpile model of active-absorbing phase transitions. Phys. Rev. E 72, 017105 (2005)

    ADS  Article  Google Scholar 

  22. 22

    Kesten H., Sidoravicius V.: Branching random walk with catalysts. Electron. J. Probab. 8, 1–51 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23

    Kesten H., Sidoravicius V.: The spread of a rumor or infection in a moving population. Ann. Probab. 33, 2402–2462 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24

    Kesten H., Sidoravicius V.: A phase transition in a model for the spread of an infection. Ill. J. Math. 50, 547–634 (2006)

    MathSciNet  MATH  Google Scholar 

  25. 25

    Kesten H., Sidoravicius V.: A shape theorem for the spread of an infection. Ann. Math. 167, 701–766 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26

    Lee, S.B.: Comment on “Fixed-energy sandpiles belong generically to directed percolation”. Phys. Rev. Lett. 110, 159601 (2013)

  27. 27

    Levine L.: Threshold state and a conjecture of poghosyan, poghosyan, priezzhev and ruelle. Commun. Math. Phys. 335(2), 1003–1017 (2015)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  28. 28

    Levine L., Peres Y.: Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile. Potential Anal. 30(1), 1–27 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29

    Liggett T.M.: Interacting Particle Systems. Springer, New York (1985)

    Google Scholar 

  30. 30

    Manna S.S.: Large-scale simulation of avalanche cluster distribution in sand pile model. J. Stat. Phys. 59, 509–521 (1990)

    ADS  Article  Google Scholar 

  31. 31

    Manna S.S.: Two-state model of self-organized criticality. J. Phys. A Math. Gen. 24, L363–L369 (1991)

    ADS  Article  Google Scholar 

  32. 32

    Marro J., Dickman R.: Nonequilibrium Phase Transitions in Lattice Models. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  33. 33

    Rolla L.T., Sidoravicius V.: Absorbing-state phase transition for driven-dissipative stochastic dynamics on \({\mathbb{Z}}\). Invent. Math. 188, 127–150 (2012)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. 34

    Rolla L.T., Sidoravicius V.: Absorbing-state phase transition for driven-dissipative stochastic dynamics on \({\mathbb{Z}}\). Invent. Math. 188(1), 127–150 (2012)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  35. 35

    Rolla, L.T.: Activated Random Walks. Lecture Notes. arXiv:1507.04341(2015)

  36. 36

    Rolla, L.T., Tournier, L.: Sustained Activity for Biased Activated Random Walks at Arbitrarily Low Density. arXiv:1507.04732 (2015)

  37. 37

    Rossi M., Pastor-Satorras R., Vespignani A.: Universality class of absorbing phase transitions with a conserved field. Phys. Rev. Lett. 85, 1803–1806 (2000)

    ADS  Article  Google Scholar 

  38. 38

    Shellef E.: Nonfixation for activated random walks. ALEA 7, 137149 (2010)

    MathSciNet  MATH  Google Scholar 

  39. 39

    Sidoravicius, V., Teixeira, A.: Absorbing-State Transition for Stochastic Sandpiles and Activated Random Walks. arXiv:1412.7098 (2014)

  40. 40

    Stauffer, A., Taggi, L.: Critical Density of Activated Random Walks on \({\mathbb{Z}^d}\) and General Graphs. arXiv:1512.02397 (2015)

  41. 41

    Taggi, L.: Absorbing-state phase transition in biased activated random walk, Electron. J. Probab. 21, 1–15 (2016)

  42. 42

    Vespignani A., Dickman R., Muñoz M.A., Zapperi S.: Driving, conservation, and absorbing states in sandpiles. Phys. Rev. Lett. 81, 5676–5679 (1998)

    ADS  Article  Google Scholar 

  43. 43

    Vespignani A., Dickman R., Muñoz M.A., Zapperi S.: Absorbing-state phase transitions in fixed-energy sandpiles. Phys. Rev. E 62, 4564–4582 (2000)

    ADS  Article  Google Scholar 

  44. 44

    Vidigal R., Dickman R.: Asymptotic behavior of the order parameter in a stochastic sandpile. J. Stat. Phys. 118(1), 1–25 (2005)

    ADS  MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Riddhipratim Basu.

Additional information

Communicated by H. Spohn

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Basu, R., Ganguly, S. & Hoffman, C. Non-fixation for Conservative Stochastic Dynamics on the Line. Commun. Math. Phys. 358, 1151–1185 (2018). https://doi.org/10.1007/s00220-017-3059-7

Download citation