Skip to main content
SpringerLink
Log in

The Chemical Distance in Random Interlacements in the Low-Intensity Regime

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In \(\mathbb {Z}^d\) with \(d\ge 5\), we consider the time constant \(\rho _u\) associated to the chemical distance in random interlacements at low intensity \(u \ll 1\). We prove an upper bound of order \(u^{-1/2}\) and a lower bound of order \(u^{-1/2+\varepsilon }\). The upper bound agrees with the conjectured scale in which \(u^{1/2}\rho _u\) converges to a constant multiple of the Euclidean norm, as \(u\rightarrow 0\). Along the proof, we obtain a local lower bound on the chemical distance between the boundaries of two concentric boxes, which might be of independent interest. For both upper and lower bounds, the paper employs probabilistic bounds holding as \(u\rightarrow 0\); these bounds can be relevant in future studies of the low-intensity geometry.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Alexander, K., Chayes, J.T., Chayes, L.: The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation. Commun. Math. Phys. 131(1), 1–50 (1990)

    ADS  MathSciNet  MATH  Google Scholar 

  2. Alves, O.S.M., Machado, F.P., Popov, SYu.: The shape theorem for the frog model. Ann. Appl. Probab. 12(2), 533–546 (2002)

    MathSciNet  MATH  Google Scholar 

  3. Andres, S., Prévost, A.: First passage percolation with long-range correlations and applications to random Schrödinger operators. arXiv:2112.12096

  4. Asselah, A., Schapira, B.: Large deviations for intersections of random walks. arXiv:2005.02735

  5. Asselah, A., Schapira, B.: Deviations for the capacity of the range of a random walk. Electron. J. Probab. 25, 1–28 (2020)

    MathSciNet  MATH  Google Scholar 

  6. Asselah, A., Schapira, B., Sousi, P.: Capacity of the range of random walk on \(\mathbb{Z} ^d\). Trans. Am. Math. Soc. 370(11), 7627–7645 (2018)

    MATH  Google Scholar 

  7. Auffinger, A., Damron, M., Hanson, J.: 50 Years of First-Passage Percolation, volume 68 of University Lecture Series. American Mathematical Society, Providence (2017)

    MATH  Google Scholar 

  8. Barlow, M.T.: Random Walks and Heat Kernels on Graphs. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2017)

    Google Scholar 

  9. Benjamini, I.: Euclidean versus graph metric. In: Lovász, L., Ruzsa, I.Z., Sós, V.T. (eds.) Erdős Centennial, pp. 35–57. Springer, Berlin (2013)

    Google Scholar 

  10. Benjamini, I., Sznitman, A.S.: Giant component and vacant set for random walk on a discrete torus. J. Eur. Math. Soc. 10(1), 133–172 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Berestycki, N., Yadin, A.: Condensation of a self-attracting random walk. Ann. Inst. H. Poincaré Probab. Stat. 55(2), 835–861 (2019)

    MathSciNet  MATH  Google Scholar 

  12. Biskup, M., Louidor, O., Procaccia, E.B., Rosenthal, R.: Isoperimetry in two-dimensional percolation. Commun. Pure Appl. Math. 68(9), 1483–1531 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Biskup, M., Procaccia, E.B.: Eigenvalue versus perimeter in a shape theorem for self-interacting random walks. Ann. Appl. Probab. 28(1), 340–377 (2018)

    MathSciNet  MATH  Google Scholar 

  14. Boivin, D.: First passage percolation: the stationary case. Probab. Theory Relat. Fields 86(4), 491–499 (1990)

    MathSciNet  MATH  Google Scholar 

  15. Cai, Z., Han, X., Ye, J., Zhang, Y.: On chemical distance and local uniqueness of a sufficiently supercritical finitary random interlacement. J. Theor. Probab. (2022)

  16. Cerf, R.: The Wulff crystal in Ising and percolation models: Ecole D’Eté de Probabilités de Saint-Flour XXXIV-2004. Springer, Berlin (2006)

    MATH  Google Scholar 

  17. Cerf, R., Pisztora, A.: On the Wulff crystal in the Ising model. Ann. Probab. 28, 947–1017 (2000)

    MathSciNet  MATH  Google Scholar 

  18. Černý, J., Popov, S.: On the internal distance in the interlacement set. Electron. J. Probab. 17, 1–25 (2012)

    MathSciNet  MATH  Google Scholar 

  19. Cox, J.T., Durrett, R.: Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9(4), 583–603 (1981)

    MathSciNet  MATH  Google Scholar 

  20. Dobrushin, R.L., Koteckỳ, R., Shlosman, S.: Wulff Construction: A Global Shape from Local Interaction, vol. 104. American Mathematical Society, Providence (1992)

    MATH  Google Scholar 

  21. Drewitz, A., Ráth, B., Sapozhnikov, A.: An Introduction to Random Interlacements. Springer, Berlin (2014)

    MATH  Google Scholar 

  22. Drewitz, A., Ráth, B., Sapozhnikov, A.: On chemical distances and shape theorems in percolation models with long-range correlations. J. Math. Phys. 55(8), 083307 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  23. Duminil-Copin, H.: Limit of the Wulff crystal when approaching criticality for site percolation on the triangular lattice. Electron. Commun. Probab. 18, 1–9 (2013)

    MathSciNet  MATH  Google Scholar 

  24. Erhard, D., Poisat, J.: Asymptotics of the critical time in Wiener sausage percolation with a small radius. ALEA Lat. Am. J. Probab. Math. Stat. 13, 417–445 (2016)

    MathSciNet  MATH  Google Scholar 

  25. Gold, J.: Isoperimetry in supercritical bond percolation in dimensions three and higher. Ann. Inst. H. Poincaré Probab. Stat. 54(4), 2092–2158 (2018)

    MathSciNet  MATH  Google Scholar 

  26. Grimmett, G.R.: Percolation. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (1999)

    Google Scholar 

  27. Holmes, M., Salisbury, T.S.: A shape theorem for the orthant model. Ann. Probab. 49(3), 1237–1256 (2021)

    MathSciNet  MATH  Google Scholar 

  28. Ioffe, D., Schonmann, R.H.: Dobrushin–Koteckỳ–Shlosman theorem up to the critical temperature. Commun. Math. Phys. 199(1), 117–167 (1998)

    ADS  MATH  Google Scholar 

  29. Jain, N.C., Orey, S.: On the range of random walk. Israel J. Math. 6, 373–380 (1968)

    MathSciNet  MATH  Google Scholar 

  30. Kesten, H.: On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3(2), 296–338 (1993)

    ADS  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  32. Lawler, G.F.: Intersections of random walks. Probability and its Applications. Birkhäuser Boston, Inc., Boston (1991)

    MATH  Google Scholar 

  33. Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  34. Popov, S., Teixeira, A.: Soft local times and decoupling of random interlacements. J. Eur. Math. 17(10), 2545–2593 (2015)

    MathSciNet  MATH  Google Scholar 

  35. Procaccia, E.B., Rosenthal, R., Sapozhnikov, A.: Quenched invariance principle for simple random walk on clusters in correlated percolation models. Probab. Theory Relat. Fields 166, 619–657 (2016)

    MathSciNet  MATH  Google Scholar 

  36. Procaccia, E.B., Shellef, E.: On the range of a random walk in a torus and random interlacements. Ann. Probab. 42(4), 1590–1634 (2014)

    MathSciNet  MATH  Google Scholar 

  37. Procaccia, E.B., Tykesson, J.: Geometry of the random interlacement. Electron. Commun. Probab. 16, 528–544 (2011)

    MathSciNet  MATH  Google Scholar 

  38. Ráth, B., Sapozhnikov, A.: On the transience of random interlacements. Electron. Commun. Probab. 16, 379–391 (2011)

    MathSciNet  MATH  Google Scholar 

  39. Ráth, B., Sapozhnikov, A.: Connectivity properties of random interlacement and intersection of random walks. ALEA Lat. Am. J. Probab. Math. Stat. 9, 67–83 (2012)

    MathSciNet  MATH  Google Scholar 

  40. Ráth, B., Sapozhnikov, A.: The effect of small quenched noise on connectivity properties of random interlacements. Electron. J. Probab. 18, 1–20 (2013)

    MathSciNet  MATH  Google Scholar 

  41. Richardson, D.: Random growth in a tessellation. Math. Proc. Camb. Philos. Soc. 74(3), 515–528 (1973)

    ADS  MathSciNet  MATH  Google Scholar 

  42. Sapozhnikov, A.: Random walks on infinite percolation clusters in models with long-range correlations. Ann. Probab. 45(3), 1842–1898 (2017)

    MathSciNet  MATH  Google Scholar 

  43. Schapira, B.: Capacity of the range in dimension \(5\). Ann. Probab. 48(6), 2988–3040 (2020)

    MathSciNet  MATH  Google Scholar 

  44. Shiraishi, D.: Growth exponent for loop-erased random walk in three dimensions. Ann. Probab. 46(2), 687–774 (2018)

    MathSciNet  MATH  Google Scholar 

  45. Sznitman, A.S.: Vacant set of random interlacements and percolation. Ann. Math. 171(3), 2039–2087 (2010)

    MathSciNet  MATH  Google Scholar 

  46. VanDer Hofstad, R.: Random Graphs and Complex Networks, vol. 43. Cambridge University Press, Cambridge (2017)

    Google Scholar 

  47. Vershynin, R.: High-Dimensional Probability: An Introduction with Applications in Data Science. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2018)

    MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous referee, whose extremely dedicated and thorough work has greatly improved this paper. We would like to thank Balázs Ráth for insightful discussions that were greatly instrumental for the proof of Theorem 1. We would like to thank Artëm Sapozhnikov for insightful discussions that were greatly instrumental for the proof of Theorem 2, and for typing some of the proofs in Sect. 3.3.

Author information

Authors and Affiliations

  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Saraí Hernández-Torres

  2. Technion – Israel Institute of Technology, Haifa, Israel

    Eviatar B. Procaccia & Ron Rosenthal

Authors
  1. Saraí Hernández-Torres
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eviatar B. Procaccia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ron Rosenthal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eviatar B. Procaccia.

Additional information

Communicated by J. Ding.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

SHT is supported by ISF grant 1692/17. EBP is supported by BSF Grant 2018330 and NSF Grant DMS-1812009. RR is supported by BSF grant 2018330 and ISF grant 771/17.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hernández-Torres, S., Procaccia, E.B. & Rosenthal, R. The Chemical Distance in Random Interlacements in the Low-Intensity Regime. Commun. Math. Phys. 400, 1697–1737 (2023). https://doi.org/10.1007/s00220-023-04634-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-023-04634-8

Search

Navigation