Electronic Communications in Probability

Upper bound on the expected size of the intrinsic ball

Artem Sapozhnikov

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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

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

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

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Zentralblatt MATH identifier

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

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

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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

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  • 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.