Abstract
We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point. The behavior of this process is well understood when the radii are uniformly bounded from above. In this article, we investigate this process for unbounded (and possibly heavy tailed) radii distributions. Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter \(\lambda _c\). Moreover,
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For \(\lambda < \lambda _c\), the vacant set has a unique unbounded connected component and we give precise bounds on the one-arm probability for the occupied set, depending on the radius distribution.
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At criticality, we establish the box-crossing property, implying that no unbounded component can be found, neither in the occupied nor the vacant sets. We provide a polynomial decay for the probability of the one-arm events, under sharp conditions on the distribution of the radius.
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For \(\lambda > \lambda _c\), the occupied set has a unique unbounded component and we prove that the one-arm probability for the vacant decays exponentially fast.
The techniques we develop in this article can be applied to other models such as the Poisson Voronoi and confetti percolation.
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Ahlberg, D., Broman, E., Griffiths, S., Morris, R.: Noise sensitivity in continuum percolation. Israel J. Math. 201(2), 847–899 (2014)
Aizenman, M., Chayes, J.T., Chayes, L., Fröhlich, J., Russo, L.: On a sharp transition from area law to perimeter law in a system of random surfaces. Commun. Math. Phys. 92(1), 19–69 (1983)
Ahlberg, D., Griffiths, S., Morris, R., Tassion, V.: Quenched Voronoi percolation. Adv. Math. 286, 889–911 (2016)
Alexander, K.S.: The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees. Ann. Appl. Probab. 6(2), 466–494 (1996)
Ahlberg, D., Tykesson, J.: The Poisson Boolean model in a random scenery. In preparation
Ahlberg, D., Tassion, V., Teixeira, A.: Existence of an unbounded vacant set in subcritical continuum percolation. In preparation
Beffara, V., Duminil-Copin, H.: The self-dual point of the two-dimensional random-cluster model is critical for \(q\ge 1\). Probab. Theory Relat. Fields 153(3–4), 511–542 (2012)
Broadbent, S.R., Hammersley, J.M.: Percolation processes. I. Crystals and mazes. Proc. Camb. Philos. Soc. 53, 629–641 (1957)
Bollobás, B., Riordan, O.: The critical probability for random Voronoi percolation in the plane is 1/2. Probab. Theory Relat. Fields 136(3), 417–468 (2006)
Bollobás, B., Riordan, O.: Percolation. Cambridge University Press, New York (2006)
Benjamini, I., Schramm, O.: Exceptional planes of percolation. Probab. Theory Relat. Fields 111(4), 551–564 (1998)
Bollobás, B., Thomason, A.: Threshold functions. Combinatorica 7(1), 35–38 (1987)
Duminil-Copin, H., Sidoravicius, V., Tassion, V.: Continuity of the phase transition for planar random-cluster and Potts models with \(1\le q\le 4\). arXiv:1505.04159, To appear in Commun. Math Phys. (2015)
Erdős, P., Rényi, A.: On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5, 17–61 (1960)
Gilbert, E.N.: Random plane networks. J. Soc. Ind. Appl. Math. 9, 533–543 (1961)
Gouéré, J.-B.: Subcritical regimes in the Poisson Boolean model of continuum percolation. Ann. Probab. 36(4), 1209–1220 (2008)
Gouéré, J.-B.: Percolation in a multiscale Boolean model. ALEA Lat. Am. J Probab. Math. Stat. 11(1), 281–297 (2014)
Grimmett, G.: Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin, second edition (1999)
Garban, C., Steif, J.E.: Noise Sensitivity of Boolean Functions and Percolation. Cambridge University Press, Cambridge (2014)
Hall, P.: On continuum percolation. Ann. Probab. 13(4), 1250–1266 (1985)
Hirsch, C.: A Harris–Kesten theorem for confetti percolation. Random Struct. Algorithms 47(2), 361–385 (2015)
Jeulin, D.: Dead leaves models: from space tessellation to random functions. In: Proceedings of the International Symposium on Advances in Theory and Applications of Random Sets (Fontainebleau, 1996), pp. 137–156. World Scientific Publishing, River Edge, NJ, (1997)
Kesten, H.: The critical probability of bond percolation on the square lattice equals \({1\over 2}\). Commun. Math. Phys. 74(1), 41–59 (1980)
Kesten, H.: Percolation Theory for Mathematicians, Progress in Probability and Statistics, vol. 2. Birkhäuser Boston, Cambridge (1982)
Kahn, J., Kalai, G., Linial, N.: The influence of variables on Boolean functions. In: 29th Annual Symposium on Foundations of Computer Science, pp. 68–80 (1988)
Meester, R., Roy, R.: Uniqueness of unbounded occupied and vacant components in Boolean models. Ann. Appl. Probab. 4(3), 933–951 (1994)
Meester, R., Roy, R.: Continuum Percolation, Cambridge Tracts in Mathematics, vol. 119. Cambridge University Press, Cambridge (1996)
Meester, R., Roy, R., Sarkar, A.: Nonuniversality and continuity of the critical covered volume fraction in continuum percolation. J. Stat. Phys. 75(1–2), 123–134 (1994)
Men’shikov, M.V., Sidorenko, A.F.: Coincidence of critical points in Poisson percolation models. Teor. Veroyatnost. i Primenen. 32(3), 603–606 (1987)
Müller, T.: The critical parameter for confetti percolation equals \(1/2\). Random Struct. Algorithms. To appear
O’Donnell, R.: Analysis of Boolean functions. Cambridge University Press, Cambridge (2014)
Roy, R.: The Russo-Seymour-Welsh theorem and the equality of critical densities and the “dual” critical densities for continuum percolation on \({ R}^2\). Ann. Probab. 18(4), 1563–1575 (1990)
Roy, R.: Percolation of Poisson sticks on the plane. Probab. Theory Relat. Fields 89(4), 503–517 (1991)
Russo, L.: A note on percolation. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 43(1), 39–48 (1978)
Russo, L.: On the critical percolation probabilities. Z. Wahrsch. Verw. Gebiete 56(2), 229–237 (1981)
Russo, L.: An approximate zero–one law. Z. Wahrsch. Verw. Gebiete 61(1), 129–139 (1982)
Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 (2000)
Smirnov, S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333(3), 239–244 (2001)
Seymour, P.D., Welsh, D.J.A.: Percolation probabilities on the square lattice. Ann. Discrete Math., 3, 227–245 (1978). Advances in graph theory (Cambridge Combinatorial Conferences, Trinity College, Cambridge, 1977)
Sznitman, A.-S.: On scaling limits and Brownian interlacements. Bull. Braz. Math. Soc. (NS) 44(4), 555–592 (2013)
Talagrand, M.: On Russo’s approximate zero-one law. Ann. Probab. 22(3), 1576–1587 (1994)
Tassion, V.: Crossing probabilities for voronoi percolation. Ann. Probab. 44(5), 3385–3398, 09 (2016)
Tykesson, J., Windisch, D.: Percolation in the vacant set of Poisson cylinders. Probab. Theory Relat. Fields 154(1–2), 165–191 (2012)
Zuev, S.A., Sidorenko, A.F.: Continuous models of percolation theory. I. Teoret. Mat. Fiz. 62(1), 76–86 (1985)
Zuev, S.A., Sidorenko, A.F.: Continuous models of percolation theory. II. Teoret. Mat. Fiz. 62(2), 253–262 (1985)
Acknowledgements
We would like to thank Caio Teodoro for the careful reading, suggestions and corrections. This work began during a visit of V. T. to IMPA, that he thanks for support and hospitality. We thank the Centre Intradisciplinaire Bernoulli (CIB) and Stardû for hosting the authors. D. A. was during the course of this project financed by Grant 637-2013-7302 from the Swedish Research Council. A. T. is grateful to CNPq for its financial contribution to this work through the Grants 306348/2012-8, 478577/2012-5 and 309356/2015-6 and FAPERJ through Grant Number 202.231/2015. V. T. acknowledges support from the Swiss NSF.
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Ahlberg, D., Tassion, V. & Teixeira, A. Sharpness of the phase transition for continuum percolation in \(\mathbb {R}^2\) . Probab. Theory Relat. Fields 172, 525–581 (2018). https://doi.org/10.1007/s00440-017-0815-8
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DOI: https://doi.org/10.1007/s00440-017-0815-8