Skip to main content
Log in

Recurrence and density decay for diffusion-limited annihilating systems

  • Published:
Probability Theory and Related Fields Aims and scope Submit manuscript

Abstract

We study an infinite system of moving particles, where each particle is of type A or B. Particles perform independent random walks at rates \(D_A > 0\) and \(D_B \geqslant 0\), and the interaction is given by mutual annihilation \(A+B \rightarrow \emptyset \). The initial condition is i.i.d. with finite first moment. We show that this system is site-recurrent, that is, each site is visited infinitely many times. We also generalize a lower bound on the density decay of Bramson and Lebowitz by considering a construction that handles different jump rates.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Models with mutual annihilation were originally introduced in chemical physics for the study of radiation-chemical processes in polymers. It was proposed in [19] that a radical may move along a polymer chain, and that the act of recombination takes place when two migrating radicals encounter one another. A diffusive mechanism for the motion of radicals was proposed in [22], and a kinetic equation describing concentration of free radicals as function of the time was derived. The same model served as a prototype of multi-type diffusion-limited chemical reactions with annihilation or inert compound outcome [10, 23], and as caricature modeling particle–antiparticle annihilation of superheavy magnetic monopoles in the very early universe [26].

    Our motivation comes from the study of driven-dissipative lattice gases which undergo absorbing-state phase transitions. The authors arrived to the present model as a caricature of a system starting from an active state with critical density [12, 24]. The A-particles should correspond to regions that are slightly supercritical due to fluctuations, whereas B-particles represent slightly subcritical regions. Surprisingly enough, some of the techniques developed in this paper have been applied with success in the study of the original model [11].

  2. In multi-type systems particles are of types \(A_1,A_2,\dots ,A_M\), and jump at rate 1 according to a generously transitive transition kernel \(p(\cdot ,\cdot )\). Interaction is given by \(A_i+A_j \rightarrow \emptyset \) for any \(i \ne j\). Each site initially contains one particle of type \(A_i\) with probability \(\frac{p}{M}\) and no particles with probability \(1-p\), independently of other sites. In the one-type system the interaction is given by \(A+A\rightarrow \emptyset \) and the initial condition is i.i.d. Bernoulli. The proofs given in the next sections for site recurrence work in these settings. For the one-type system, there is a simpler and more general proof in [4].

  3. An event is “local” if its occurrence is determined by \(\big (\xi _t(x)\big )_{|x|\leqslant M,t\leqslant M}\) for some \(M<\infty \).

  4. A graph G being generously transitive is stronger than being unimodular, and it is neither stronger nor weaker than being Cayley. An example of a graph that is generously transitive but not Cayley is the product \(P \times \mathbb {Z}\), where P is the Petersen graph. Cayley graphs of Abelian groups are generously transitive. An example of a graph that is not generously transitive but is Cayley is the free product \(\mathbb {Z}_2 * \mathbb {Z}_3\).

References

  1. Adelman, O.: Some use of some “symmetries” of some random process. Ann. Inst. H. Poincaré Sect. B (N. S.) 12, 193–197 (1976)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Arratia, R.: Site recurrence for annihilating random walks on \({ Z}_{d}\). Ann. Probab. 11, 706–713 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benjamini, I., Foxall, E., Gurel-Gurevich, O., Junge, M., Kesten, H.: Site recurrence for coalescing random walk. Electron. Commun. Probab. 21, 47 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bramson, M., Lebowitz, J.L.: Asymptotic behavior of densities in diffusion-dominated annihilation reactions. Phys. Rev. Lett. 61, 2397–2400 (1988)

    Article  Google Scholar 

  6. Bramson, M., Lebowitz, J.L.: Asymptotic behavior of densities in diffusion dominated two-particle reactions. Phys. A 168, 88–94 (1990)

    Article  MathSciNet  Google Scholar 

  7. Bramson, M., Lebowitz, J.L.: Asymptotic behavior of densities for two-particle annihilating random walks. J. Stat. Phys. 62, 297–372 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bramson, M., Lebowitz, J.L.: Spatial structure in diffusion-limited two-particle reactions. J. Stat. Phys. 65, 941–951 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bramson, M., Lebowitz, J.L.: Spatial structure in low dimensions for diffusion limited two-particle reactions. Ann. Appl. Probab. 11, 121–181 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Balagurov, B.V., Vaks, V.G.: Random walks of a particle on lattices with traps. Zh. Eksp. Teor. Fiz. 65, 1939–1946 (1973)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Erdős, P., Ney, P.: Some problems on random intervals and annihilating particles. Ann. Probab. 2, 828–839 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  14. Griffeath, D.: Annihilating and coalescing random walks on \({\bf Z}_{d}\). Z. Wahrsch. Verw. Geb. 46, 55–65 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  15. Griffeath, D.: Additive and Cancellative Interacting Particle Systems, Vol. 724 of Lecture Notes in Mathematics. Springer, Berlin (1979)

    Book  Google Scholar 

  16. Holley, R.A., Liggett, T.M.: Ergodic theorems for weakly interacting infinite systems and the voter model. Ann. Probab. 3, 643–663 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  17. Holley, R., Stroock, D.: Dual processes and their application to infinite interacting systems. Adv. Math. 32, 149–174 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hughes, B.D.: Random Walks and Random Environments. Random Walks, vol. 1. The Clarendon Press/ Oxford University Press, New York (1995)

  19. Koritskii, A., Molin, I., Shamshev, V., Buben, N., Voevodskii, V.: An electronic paramagnetic resonance study of the radicals formed in fast electron irradiation of polyethylene. Polym. Sci. USSR 1, 458–472 (1960)

    Article  Google Scholar 

  20. Lootgieter, J.-C.: Problèmes de récurrence concernant des mouvements aléatoires de particules sur \({\bf Z}\) avec destruction. Ann. Inst. H. Poincaré Sect. B (N. S.) 13, 127–139 (1977)

    MathSciNet  MATH  Google Scholar 

  21. Lyons, R., Peres, Y.: Probability on Trees and Networks. Book in Preparation. Current version, http://mypage.iu.edu/~rdlyons

  22. Ovchinnikov, A.A., Belyi, A.A.: The kinetics of the destruction of radicals in polymers. Theor. Exp. Chem. 2, 405–408 (1968)

    Article  Google Scholar 

  23. Ovchinnikov, A., Zeldovich, Y.: Role of density fluctuations in bimolecular reaction kinetics. Chem. Phys. 28, 215–218 (1978)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Schwartz, D.: On hitting probabilities for an annihilating particle model. Ann. Probab. 6, 398–403 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  26. Toussaint, D., Wilczek, F.: Particle–antiparticle annihilation in diffusive motion. J. Chem. Phys. 78, 2642 (1983)

    Article  Google Scholar 

Download references

Acknowledgements

We thank E. Andjel for fruitful discussions. M.C. and V.S thank MSRI for hospitality and support. This project was supported by grants Programa Iniciativa Científica Milenio grant number NC130062 through Nucleus Millennium Stochastic Models of Complex and Disordered Systems, PIP 11220130100521CO, PICT-2015-3154, PICT-2013-2137, PICT-2012-2744, Conicet-45955 and MinCyT-BR-13/14.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. Sidoravicius.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cabezas, M., Rolla, L.T. & Sidoravicius, V. Recurrence and density decay for diffusion-limited annihilating systems. Probab. Theory Relat. Fields 170, 587–615 (2018). https://doi.org/10.1007/s00440-017-0763-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-017-0763-3

Mathematics Subject Classification

Navigation