Electronic Communications in Probability

Critical Exponents in Percolation via Lattice Animals

Alan Hammond

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Abstract

We examine the percolation model on $\mathbb{Z}^d$ by an approach involving lattice animals and their surface-area-to-volume ratio. For $\beta \in [0,2(d-1))$, let $f(\beta)$ be the asymptotic exponential rate in the number of edges of the number of lattice animals containing the origin which have surface-area-to-volume ratio $\beta$. The function $f$ is bounded above by a function which may be written in an explicit form. For low values of $\beta$ ($\beta \leq 1/p_c - 1$), equality holds, as originally demonstrated by F. Delyon. For higher values ($\beta > 1/p_c - 1$), the inequality is strict.

We introduce two critical exponents, one of which describes how quickly $f$ falls away from the explicit form as $\beta$ rises from $1/p_c - 1$, and the second of which describes how large clusters appear in the marginally subcritical regime of the percolation model. We demonstrate that the pair of exponents must satisfy certain inequalities. Other such inequalities yield sufficient conditions for the absence of an infinite cluster at the critical value (c.f. [4]). The first exponent is related to one of a more conventional nature in the scaling theory of percolation, that of correlation size. In deriving this relation, we find that there are two possible behaviours, depending on the value of the first exponent, for the typical surface-area-to-volume ratio of an unusually large cluster in the marginally subcritical regime.

Article information

Source
Electron. Commun. Probab., Volume 10 (2005), paper no. 6, 45-59.

Dates
Accepted: 4 March 2005
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465058071

Digital Object Identifier
doi:10.1214/ECP.v10-1131

Mathematical Reviews number (MathSciNet)
MR2119153

Zentralblatt MATH identifier
1060.60094

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

Citation

Hammond, Alan. Critical Exponents in Percolation via Lattice Animals. Electron. Commun. Probab. 10 (2005), paper no. 6, 45--59. doi:10.1214/ECP.v10-1131. https://projecteuclid.org/euclid.ecp/1465058071


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References

  • Michael Aizenman and Charles M. Newman. Tree graph inequalities and critical behavior in percolation models. J. Statist. Phys 36(1-2):107-143, 1984.
  • F. Delyon. Taille, forme et nombre des amas dans les problemes de percolation. These de 3eme cycle, Universite Pierre et Marie Curie, Paris, 1980.
  • A. Telcs. Random walks on graphs, electric networks and fractals. Prob. Th. Rel. Fields 82 (1989), 435-451.
  • S. Flesia, D.S. Gaunt, C.E. Soteros and S.G.Whittington. Statistics of collapsing lattice animals. J. Phys. A 27(17): 5831-5846, 1991.
  • Alan Hammond. A lattice animal approach to percolation. J. Phys. A 27(17): 5831-5846, 1991.
  • Harry Kesten and Yu Zhang. The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Probab. 18 (2): 537-555, 1990.
  • Neal Madras. A rigorous bound on the critical exponent for the number of lattice tress, animals, and polygons. J. Statist. Phys. 78(3-4): 681-699, 1995..