Some Sufficient Conditions for Infinite Collisions of Simple Random Walks on a Wedge Comb

Xinxing Chen (Shanghai Jiaotong University)
Dayue Chen (Peking University)

Abstract


In this paper, we give some sufficient conditions for the infinite collisions of independent simple random walks on a wedge comb with profile $\{f(n):n\in\mathbb{Z}\}$. One interesting result is that two independent simple random walks on the wedge comb will collide infinitely many times if $f(n)$ has a growth order as $n\log(n)$. On the other hand, if $\{f(n):n\in\mathbb{Z}\}$ are given by i.i.d. non-negative random variables with finite mean, then for almost all wedge combs with such profile, three independent simple random walks on it will collide infinitely many times

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1341-1355

Publication Date: August 9, 2011

DOI: 10.1214/EJP.v16-907

References

  1. Barlow, M.T., Peres, Y., Sousi, P., Collisions of Random Walks,http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.3255v1.pdf.
  2. Chen, XinXing; Chen, DaYue. Two random walks on the open cluster of $\Bbb Z\sp 2$ meet infinitely often. Sci. China Math. 53 (2010), no. 8, 1971--1978. MR2679079
  3. Chen, Dayue; Wei, Bei; Zhang, Fuxi. A note on the finite collision property of random walks. Statist. Probab. Lett. 78 (2008), no. 13, 1742--1747. MR2453913 (2009k:60024)
  4. Durrett, Richard. Probability.Theory and examples.The Wadsworth & Brooks/Cole Statistics/Probability Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991. x+453 pp. ISBN: 0-534-13206-5 MR1068527 (91m:60002)
  5. Krishnapur, Manjunath; Peres, Yuval. Recurrent graphs where two independent random walks collide finitely often. Electron. Comm. Probab. 9 (2004), 72--81 (electronic). MR2081461 (2005h:60017)
  6. Levin, David A.; Peres, Yuval; Wilmer, Elizabeth L. Markov chains and mixing times.With a chapter by James G. Propp and David B. Wilson.American Mathematical Society, Providence, RI, 2009. xviii+371 pp. ISBN: 978-0-8218-4739-8 MR2466937 (2010c:60209)
  7. Liggett, Thomas M. Interacting particle systems.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 276. Springer-Verlag, New York, 1985. xv+488 pp. ISBN: 0-387-96069-4 MR0776231 (86e:60089)
  8. Pólya, George. Collected papers. Vol. IV.Probability; combinatorics; teaching and learning in mathematics.Edited by Gian-Carlo Rota, M. C. Reynolds and R. M. Shortt.Mathematicians of Our Time, 22. MIT Press, Cambridge, MA, 1984. ix+642 pp. ISBN: 0-262-16097-8 MR0758990 (85m:01108b)
  9. Revesz, P., Random walk in random and non-random environments. 2nd edition, World Scientific Publishing Co., New Jersey, 2005. \renewcommand 2168855
  10. Woess, Wolfgang. Random walks on infinite graphs and groups.Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge, 2000. xii+334 pp. ISBN: 0-521-55292-3 MR1743100 (2001k:60006)
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.