Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Exact value of the resistance exponent for four dimensional random walk trace
Download PDF
Download PDF
  • Published: 15 February 2011

Exact value of the resistance exponent for four dimensional random walk trace

  • Daisuke Shiraishi1 

Probability Theory and Related Fields volume 153, pages 191–232 (2012)Cite this article

  • 227 Accesses

  • 6 Citations

  • Metrics details

A Correction to this article was published on 05 November 2022

This article has been updated

Abstract

Let S be a simple random walk starting at the origin in \({\mathbb{Z}^{4}}\). We consider \({{\mathcal G}=S[0,\infty)}\) to be a random subgraph of the integer lattice and assume that a resistance of unit 1 is put on each edge of the graph \({{\mathcal G}}\). Let \({R_{{\mathcal G}}(0,S_{n})}\) be the effective resistance between the origin and S n . We derive the exact value of the resistance exponent; more precisely, we prove that \({n^{-1}E(R_{{\mathcal G}}(0,S_{n}))\approx (\log n)^{-\frac{1}{2}}}\). As an application, we obtain sharp heat kernel estimates for random walk on \({\mathcal G}\) at the quenched level. These results give the answer to the problem raised by Burdzy and Lawler (J Phys A Math Gen 23:L23–L28, 1990) in four dimensions.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

Change history

  • 05 November 2022

    A Correction to this paper has been published: https://doi.org/10.1007/s00440-022-01160-x

References

  1. Burdzy K., Lawler G.F.: Rigorous exponent inequalities for random walks. J. Phys. A Math. Gen. 23(1), L23–L28 (1990)

    Article  MathSciNet  Google Scholar 

  2. Dvoretzky, A., Erdos, P.: Some problems on random walk in space. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950. pp. 353–367. University of California Press, Berkeley and Los Angeles (1951)

  3. Hamana Y.: An almost sure invariance principle for the range of random walks. Stochast. Process. Appl. 78(2), 131–143 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kumagai T., Misumi J.: Heat kernel estimates for strongly recurrent random walk on random media. J. Theor. Probab. 21(4), 910–935 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lawler G.F.: The probability of intersection of independent random walks in four dimensions. Commun. Math. Phys. 86(4), 539–554 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lawler, G.F.: Intersections of random walks. Probability and its Applications. Birkhauser Boston, Inc., Boston, MA (1991). (soft-cover version)

  7. Lawler G.F.: Escape probabilities for slowly recurrent sets. Probab. Theory Relat. Fields 94, 91–117 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Manna S.S., Guttmann A.J., Hughes B.D.: Diffusion on two-dimensional random walks. Phys. Rev. A (3) 39(8), 4337–4340 (1989)

    Article  MathSciNet  Google Scholar 

  9. Shiraishi D.: Heat kernel for random walk trace on \({\mathbb{Z}^{3}}\) and \({\mathbb{Z}^{4}}\). Ann. Inst. H. Poincaré Probab. Statist. 46(4), 1001–1024 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, Japan

    Daisuke Shiraishi

Authors
  1. Daisuke Shiraishi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daisuke Shiraishi.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Shiraishi, D. Exact value of the resistance exponent for four dimensional random walk trace. Probab. Theory Relat. Fields 153, 191–232 (2012). https://doi.org/10.1007/s00440-011-0343-x

Download citation

  • Received: 04 December 2009

  • Revised: 11 January 2011

  • Published: 15 February 2011

  • Issue Date: June 2012

  • DOI: https://doi.org/10.1007/s00440-011-0343-x

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification (2000)

  • 82B41
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature