Electronic Communications in Probability

On the size of the largest cluster in 2D critical percolation

Jacob van den Berg and Rene Conijn

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Abstract

We consider (near-)critical percolation on the square lattice. Let $\mathcal{M}_{n}$ be the size of the largest open cluster contained in the box $[-n,n]^2$, and let $\pi(n)$ be the probability that there is an open path from $O$ to the boundary of the box. It is well-known that for all $0< a < b$ the probability that $\mathcal{M}_{n}$ is smaller than $a n^2 \pi(n)$ and the probability that $\mathcal{M}_{n}$ is larger than $b n^2 \pi(n)$ are bounded away from $0$ as $n \rightarrow \infty$. It is a natural question, which arises for instance in the study of so-called frozen-percolation processes, if a similar result holds for the probability that $\mathcal{M}_{n}$ is {\em between} $a n^2 \pi(n)$ and $b n^2 \pi(n)$. By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is affirmative. The `sublinearity' of $1/\pi(n)$ appears to be essential for the argument.<br />

Article information

Source
Electron. Commun. Probab., Volume 17 (2012), paper no. 58, 13 pp.

Dates
Accepted: 12 December 2012
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465263191

Digital Object Identifier
doi:10.1214/ECP.v17-2263

Mathematical Reviews number (MathSciNet)
MR3005731

Zentralblatt MATH identifier
1320.60161

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60C05: Combinatorial probability

Keywords
Critical percolation Cluster size

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

van den Berg, Jacob; Conijn, Rene. On the size of the largest cluster in 2D critical percolation. Electron. Commun. Probab. 17 (2012), paper no. 58, 13 pp. doi:10.1214/ECP.v17-2263. https://projecteuclid.org/euclid.ecp/1465263191


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References

  • Borgs, C.; Chayes, J. T.; Kesten, H.; Spencer, J. Uniform boundedness of critical crossing probabilities implies hyperscaling. Statistical physics methods in discrete probability, combinatorics, and theoretical computer science (Princeton, NJ, 1997). Random Structures Algorithms 15 (1999), no. 3-4, 368–413.
  • Borgs, C.; Chayes, J. T.; Kesten, H.; Spencer, J. The birth of the infinite cluster: finite-size scaling in percolation. Dedicated to Joel L. Lebowitz. Comm. Math. Phys. 224 (2001), no. 1, 153–204.
  • Grimmett, Geoffrey. Percolation. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6
  • Járai, Antal A. Incipient infinite percolation clusters in 2D. Ann. Probab. 31 (2003), no. 1, 444–485.
  • Kesten, Harry. The incipient infinite cluster in two-dimensional percolation. Probab. Theory Related Fields 73 (1986), no. 3, 369–394.
  • Kesten, Harry. Scaling relations for $2$D-percolation. Comm. Math. Phys. 109 (1987), no. 1, 109–156.
  • Nolin, Pierre. Near-critical percolation in two dimensions. Electron. J. Probab. 13 (2008), no. 55, 1562–1623.
  • van den Berg, J.; Kesten, H. Inequalities with applications to percolation and reliability. J. Appl. Probab. 22 (1985), no. 3, 556–569.
  • van den Berg, Jacob; de Lima, Bernardo N. B.; Nolin, Pierre. A percolation process on the square lattice where large finite clusters are frozen. Random Structures Algorithms 40 (2012), no. 2, 220–226.