## Electronic Communications in Probability

### Boundary rules and breaking of self-organized criticality in 2D frozen percolation

#### Abstract

We study frozen percolation on the (planar) triangular lattice, where connected components stop growing (“freeze”) as soon as their “size” becomes at least $N$, for some parameter $N \geq 1$. The size of a connected component can be measured in several natural ways, and we consider the two particular cases of diameter and volume (i.e. number of sites).

Diameter-frozen and volume-frozen percolation have been studied in previous works ([5, 15] and [6, 4], resp.), and they display radically different behaviors. These works adopt the rule that the boundary of a frozen cluster stays vacant forever, and we investigate the influence of these “boundary rules” in the present paper. We prove the (somewhat surprising) result that they strongly matter in the diameter case, and we discuss briefly the volume case.

#### Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 65, 15 pp.

Dates
Accepted: 2 November 2017
First available in Project Euclid: 23 November 2017

https://projecteuclid.org/euclid.ecp/1511406073

Digital Object Identifier
doi:10.1214/17-ECP98

Mathematical Reviews number (MathSciNet)
MR3734104

Zentralblatt MATH identifier
06827047

#### Citation

van den Berg, Jacob; Nolin, Pierre. Boundary rules and breaking of self-organized criticality in 2D frozen percolation. Electron. Commun. Probab. 22 (2017), paper no. 65, 15 pp. doi:10.1214/17-ECP98. https://projecteuclid.org/euclid.ecp/1511406073

#### References

• [1] David J. Aldous, The percolation process on a tree where infinite clusters are frozen, Math. Proc. Cambridge Philos. Soc. 128 (2000), no. 3, 465–477.
• [2] Per Bak, How nature works: the science of self-organized criticality, Copernicus, New York, 1996.
• [3] Jacob van den Berg and Rachel Brouwer, Self-organized forest-fires near the critical time, Comm. Math. Phys. 267 (2006), no. 1, 265–277.
• [4] Jacob van den Berg, Demeter Kiss, and Pierre Nolin, Two-dimensional volume-frozen percolation: deconcentration and prevalence of mesoscopic clusters, Ann. Sci. Éc. Norm. Supér. (4), to appear.
• [5] Jacob van den Berg, Bernardo N. B. de Lima, and Pierre Nolin, A percolation process on the square lattice where large finite clusters are frozen, Random Structures Algorithms 40 (2012), no. 2, 220–226.
• [6] Jacob van den Berg and Pierre Nolin, Two-dimensional volume-frozen percolation: exceptional scales, Ann. Appl. Probab. 27 (2017), no. 1, 91–108.
• [7] Michael Damron, Artëm Sapozhnikov, and Bálint Vágvölgyi, Relations between invasion percolation and critical percolation in two dimensions, Ann. Probab. 37 (2009), no. 6, 2297–2331.
• [8] Barbara Drossel and Franz Schwabl, Self-organized critical forest-fire model, Phys. Rev. Lett. 69 (1992), 1629–1632.
• [9] Rick Durrett, Ten lectures on particle systems, Lectures on probability theory (Saint-Flour, 1993), Lecture Notes in Math., vol. 1608, Springer, Berlin, 1995, pp. 97–201.
• [10] Christophe Garban, Gábor Pete, and Oded Schramm, The scaling limits of near-critical and dynamical percolation, J. Eur. Math. Soc., to appear.
• [11] Geoffrey Grimmett, Percolation, second ed., Grundlehren der Mathematischen Wissenschaften, vol. 321, Springer-Verlag, Berlin, 1999.
• [12] Henrik J. Jensen, Self-organized criticality: emergent complex behavior in physical and biological systems, Cambridge Lecture Notes in Physics, vol. 10, Cambridge University Press, Cambridge, 1998.
• [13] Harry Kesten, Percolation theory for mathematicians, Progress in Probability and Statistics, vol. 2, Birkhäuser, Boston, 1982.
• [14] Harry Kesten, Scaling Relations for 2D-percolation, Comm. Math. Phys. 109 (1987), 109–156.
• [15] Demeter Kiss, Frozen percolation in two dimensions, Probab. Theory Related Fields 163 (2015), nos 3–4, 713–768.
• [16] Gregory F. Lawler, Oded Schramm, and Wendelin Werner, Values of Brownian intersection exponents. I. Half-plane exponents, Acta Math. 187 (2001), no. 2, 237–273.
• [17] Gregory F. Lawler, Oded Schramm, and Wendelin Werner, Values of Brownian intersection exponents. II. Plane exponents, Acta Math. 187 (2001), no. 2, 275–308.
• [18] Gregory F. Lawler, Oded Schramm, and Wendelin Werner, One-arm exponent for critical 2D percolation, Electron. J. Probab. 7 (2002), no. 2, 13 pp.
• [19] Pierre Nolin, Near-critical percolation in two dimensions, Electron. J. Probab. 13 (2008), no. 55, 1562–1623.
• [20] Pierre Nolin, SLE(6) and the geometry of diffusion fronts, Preprint arXiv:0912.3770, 2009.
• [21] Pierre Nolin and Wendelin Werner, Asymmetry of near-critical percolation interfaces, J. Amer. Math. Soc. 22 (2009), no. 3, 797–819.
• [22] Bernard Sapoval, Michel Rosso, and Jean-François Gouyet, The fractal nature of a diffusion front and the relation to percolation, J. Physique Lett. 46 (1985), no. 4, 149–156.
• [23] Oded Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221–288.
• [24] Stanislav Smirnov, Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 3, 239–244.
• [25] Stanislav Smirnov and Wendelin Werner, Critical exponents for two-dimensional percolation, Math. Res. Lett. 8 (2001), nos 5–6, 729–744.
• [26] Walter H. Stockmayer, Theory of molecular size distribution and gel formation in branched-chain polymers, Journal of Chemical Physics 11 (1943), 45–55.
• [27] David Wilkinson and Jorge F. Willemsen, Invasion percolation: a new form of percolation theory, Journal of Physics A: Mathematical and General 16 (1983), 3365–3376.