Electronic Communications in Probability

Upper bound on the expected size of the intrinsic ball

Artem Sapozhnikov

Full-text: Open access

Abstract

We give a short proof of Theorem 1.2 (i) from the paper "The Alexander-Orbach conjecture holds in high dimensions" by G. Kozma and A. Nachmias. We show that the expected size of the intrinsic ball of radius $r$ is at most $Cr$ if the susceptibility exponent is at most 1. In particular, this result follows if the so-called triangle condition holds.

Article information

Source
Electron. Commun. Probab., Volume 15 (2010), paper no. 28, 297-298.

Dates
Accepted: 23 July 2010
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465243971

Digital Object Identifier
doi:10.1214/ECP.v15-1553

Mathematical Reviews number (MathSciNet)
MR2670196

Zentralblatt MATH identifier
1226.60135

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

Keywords
Critical percolation high-dimensional percolation triangle condition chemical distance intrinsic ball

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Sapozhnikov, Artem. Upper bound on the expected size of the intrinsic ball. Electron. Commun. Probab. 15 (2010), paper no. 28, 297--298. doi:10.1214/ECP.v15-1553. https://projecteuclid.org/euclid.ecp/1465243971


Export citation

References

  • M. Aizenman and Ch. Newman. Tree graph inequalities and critical behavior in percolation models. J. Statist. Phys. 36 (1984), 107-143.
  • G. Grimmett. Percolation. Springer-Verlag (1999) Berlin, Second edition.
  • T. Hara and G. Slade. Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys. 128 (1990), 333-391.
  • M. Heydenreich, R. van der Hofstad and A. Sakai. Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk. J. Statist. Phys. 132(6) (2008), 1001-1049.
  • G. Kozma and A. Nachmias. The Alexander-Orbach conjecture holds in high dimensions. Invent. Math. 178(3) (2009), 635-654.
  • G. Kozma. Percolation on a product of two trees. arXiv:1003.5240.
  • G. Kozma. The triangle and the open triangle. To appear in Ann. Inst. Henri Poincar'e Probab. Stat. (2010). arXiv:0907.1959.
  • R. Schonmann. Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. Commun. Math. Phys. 219(2) (2001), 271-322.
  • R. Schonmann. Mean-filed criticality for percolation on planar non-amenable graphs. Commun. Math. Phys. 225(3) (2002), 453-463.
  • C. Wu. Critical behavior of percolation and Markov fields on branching planes. J. Appl. Probab. 30(3) (1993), 538-547.