Electronic Journal of Probability

Collisions of several walkers in recurrent random environments

Alexis Devulder, Nina Gantert, and Françoise Pène

Full-text: Open access

Abstract

We consider $d$ independent walkers on $\mathbb{Z} $, $m$ of them performing simple symmetric random walk and $r= d-m$ of them performing recurrent RWRE (Sinai walk), in $I$ independent random environments. We show that the product is recurrent, almost surely, if and only if $m\leq 1$ or $m=d=2$. In the transient case with $r\geq 1$, we prove that the walkers meet infinitely often, almost surely, if and only if $m=2$ and $r \geq I= 1$. In particular, while $I$ does not have an influence for the recurrence or transience, it does play a role for the probability to have infinitely many meetings. To obtain these statements, we prove two subtle localization results for a single walker in a recurrent random environment, which are of independent interest.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 90, 34 pp.

Dates
Received: 5 December 2017
Accepted: 3 July 2018
First available in Project Euclid: 12 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1536717749

Digital Object Identifier
doi:10.1214/18-EJP192

Mathematical Reviews number (MathSciNet)
MR3858918

Zentralblatt MATH identifier
06964784

Subjects
Primary: 60K37: Processes in random environments 60G50: Sums of independent random variables; random walks

Keywords
random walk random environment collisions recurrence transience

Rights
Creative Commons Attribution 4.0 International License.

Citation

Devulder, Alexis; Gantert, Nina; Pène, Françoise. Collisions of several walkers in recurrent random environments. Electron. J. Probab. 23 (2018), paper no. 90, 34 pp. doi:10.1214/18-EJP192. https://projecteuclid.org/euclid.ejp/1536717749


Export citation

References

  • [1] Andreoletti, P.: Alternative proof for the localization of Sinai’s walk. J. Stat. Phys. 118, (2005), 883–933.
  • [2] Andreoletti, P. and Devulder, A.: Localization and number of visited valleys for a transient diffusion in random environment. Electron. J. Probab. 20, no 56, (2015), 1–58.
  • [3] Barlow, M., Peres, Y. and Sousi, P.: Collisions of random walks. Ann. Inst. H. Poincaré Probab. Stat. 48, no 4, (2012), 922–946.
  • [4] Bovier, A. and Faggionato, A.: Spectral analysis of Sinai’s walk for small eigenvalues. Ann. Probab. 36, (2008), 198–254.
  • [5] Brox, Th.: A one-dimensional diffusion process in a Wiener medium. Ann. Probab. 14, (1986), 1206–1218.
  • [6] Campari, R. and Cassi, D.: Random collisions on branched networks: How simultaneous diffusion prevents encounters in inhomogeneous structures. Physical Review E 86.2, (2012), 021110.
  • [7] Cheliotis, D.: Diffusion in random environment and the renewal theorem. Ann. Probab. 33, (2005), 1760–1781.
  • [8] Cheliotis, D.: Localization of favorite points for diffusion in a random environment. Stoch. Proc. Appl. 118, (2008), 1159–1189.
  • [9] Chung, K. L.: A course in probability theory. Academic Press, Inc., San Diego, Third edition, 2001. xviii+419 pp.
  • [10] Deheuvels, P. and Révész, P.: Simple random walk on the line in random environment. Probab. Theory Related Fields 72, (1986), 215–230.
  • [11] Dembo, A., Gantert, N., Peres, Y. and Shi, Z.: Valleys and the maximum local time for random walk in random environment. Probab. Theory Related Fields 137, (2007), 443–473.
  • [12] Devulder, A.: Persistence of some additive functionals of Sinai’s walk. Ann. Inst. H. Poincaré Probab. Stat. 52, (2016), 1076–1105.
  • [13] Devulder, A., Gantert, N. and Pène, F.: Arbitrary many walkers meet infinitely often in a subballistic random environment. In preparation, (2018+).
  • [14] Doyle, P. G. and Snell, E. J.: Probability: Random walks and Electrical Networks. Carus Math. Monographs 22, Math. Assoc. Amer., Washington DC, 1984. xiv+159 pp.
  • [15] Dvoretzky, A. and Erdös, P.: Some problems on random walk in space, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, (1951), 353–367.
  • [16] Freire, M.V.: Application of moderate deviation techniques to prove Sinai theorem on RWRE. J. Stat. Phys. 160 (2), (2015), 357–370.
  • [17] Gallesco, C.: Meeting time of independent random walks in random environment. ESAIM Probab. Stat. 17, (2013), 257–292.
  • [18] Gantert, N., Kochler M. and Pène, F.: On the recurrence of some random walks in random environment. ALEA Lat. Am. J. Probab. Math. Stat. 11, (2014), 483–502.
  • [19] Golosov, A. O.: Localization of random walks in one-dimensional random environments. Commun. Math. Phys. 92, (1984), 491–506.
  • [20] Hirsch, W. M.: A strong law for the maximum cumulative sum of independent random variables. Comm. Pure Appl. Math. 18, (1965), 109–127.
  • [21] Hu, Y. and Shi, Z.: The limits of Sinai’s simple random walk in random environment. Ann. Probab. 26, (1998), 1477–1521.
  • [22] Hu, Y. and Shi, Z.: Moderate deviations for diffusions with Brownian potentials. Ann. Probab. 32, (2004), 3191–3220.
  • [23] Kawazu, K., Tamura, Y. and Tanaka, H.: Limit theorems for one-dimensional diffusions and random walks in random environments. Probab. Theory Related Fields 80, (1989), 501–541.
  • [24] Kochen, S. P. and Stone C. J.: A note on the Borel-Cantelli lemma. Illinois J. Math. 8, (1964), 248–251.
  • [25] Komlós, J., Major, P. and Tusnády, G.: An approximation of partial sums of independent RV’s and the sample df. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32, (1975), 111–131.
  • [26] Krishnapur, M. and Peres, Y.: Recurrent graphs where two independent random walks collide finitely often. Electron. Comm. Probab. 9, (2004), 72–81.
  • [27] Lawler, G. F. and Limic, V.: Random walk: a modern introduction, Cambridge Studies in Advanced Mathematics 123, Cambridge University Press, Cambridge, 2010. xii+364 pp.
  • [28] Neveu, J.: Bases mathématiques du calcul des probabilités. Masson et Cie, Éditeurs, Paris, 1964. xiii+203 pp.
  • [29] Neveu J. and Pitman J.: Renewal property of the extrema and tree property of the excursion of a one-dimensional Brownian motion. Séminaire de Probabilités XXIII, (1989) 239–247, Lecture Notes in Math., Springer, Berlin, 1372.
  • [30] Pólya, G: Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz. Math. Ann. 84, (1921), 149–160.
  • [31] Pólya, G: Collected papers, Vol IV. Probability; combinatorics; teaching and learning in mathematics. Edited by Gian-Carlo Rota, M. C. Reynolds and R. M. Shortt. MIT Press, Cambridge, Massachusetts, 1984. ix+642 pp.
  • [32] Shi, Z. and Zindy, O.: A weakness in strong localization for Sinai’s walk. Ann. Probab. 35, (2007), 1118–1140.
  • [33] Sinai, Ya. G.: The limit behavior of a one-dimensional random walk in a random environment. Teor. Veroyatnost. i Primenen 27, (1982), 247–258.
  • [34] Solomon, F.: Random walks in a random environment. Ann. Probab. 3, (1975), 1–31.
  • [35] Spitzer, F.: Principles of random walk, Graduate Texts in Mathematics, Vol. 34, Second edition. Springer-Verlag, New York-Heidelberg, 1976. xiii+408 pp.
  • [36] Zeitouni, O.: Random walks in random environment. École d’été de probabilités de Saint-Flour 2001. Lecture Notes in Math. 1837 189–312. Springer, Berlin, 2004.