Subcritical phase of $d$-dimensional Poisson–Boolean percolation and its vacant set

Duminil-Copin, Hugo; Raoufi, Aran; Tassion, Vincent

Annales Henri Lebesgue, Tome 3 (2020), pp. 677-700.

Nous prouvons que la percolation Poisson–Booléenne sur ${ℝ}^d$ a une transition de phase rapide en toute dimension sous l’hypothèse que la distribution pour le rayon des boules a un moment d’ordre $5d-3$ qui est fini (en particulier, nous ne supposons pas que cette distribution est à support compact). Ceci représente la première preuve de ce fait en dimension $d\ge 3$ pour un modèle qui n’exhibe pas de décroissance exponentielle du rayon de la composante connexe de l’origine dans le régime sous-critique. Plus précisément, nous montrons que dans tout le régime sous-critique, l’expérance de la taille de la composante connexe de l’origine est finie, et de plus nous obtenons des estimées sur la probabilité que cette composante connexe ait un rayon plus grand que $n$ : quand la distribution pour le rayon des boules a un moment exponentiel fini, cette probabilité décroit exponentiellement vite en $n$, et quand cette distribution a une queue lourde, la probabilité devient équivalente à la probabilité qu’il existe une boule de diamètre au moins $n$ contenant l’origine. Le même résultat s’étend au complémentaire de la percolation Poisson–Booléenne (aussi appelé ensemble vacant).

We prove that the Poisson–Boolean percolation on ${ℝ}^d$ undergoes a sharp phase transition in any dimension under the assumption that the radius distribution has a $5d-3$ finite moment (in particular we do not assume that the distribution is bounded). To the best of our knowledge, this is the first proof of sharpness for a model in dimension $d\ge 3$ that does not exhibit exponential decay of connectivity probabilities in the subcritical regime. More precisely, we prove that in the whole subcritical regime, the expected size of the cluster of the origin is finite, and furthermore we obtain bounds for the origin to be connected to distance $n$: when the radius distribution has a finite exponential moment, the probability decays exponentially fast in $n$, and when the radius distribution has heavy tails, the probability is equivalent to the probability that the origin is covered by a ball going to distance $n$ (this result is new even in two dimensions). In the supercritical regime, it is proved that the probability of the origin being connected to infinity satisfies a mean-field lower bound. The same proof carries on to conclude that the vacant set of Poisson–Boolean percolation on ${ℝ}^d$ undergoes a sharp phase transition.

Classification : 82B43
Mots clés : continuum percolation, sharp threshold, phase transition, subcritical phase

[AB87] Aizenman, Michael; Barsky, David J. Sharpness of the phase transition in percolation models, Commun. Math. Phys., Volume 108 (1987) no. 3, pp. 489-526 | DOI | MR | Zbl

[ABF87] Aizenman, Michael; Barsky, David J.; Fernández, Roberto The phase transition in a general class of Ising-type models is sharp, J. Statist. Phys., Volume 47 (1987) no. 3-4, pp. 343-374 | DOI | MR

[ATT16] Ahlberg, Daniel; Tassion, Vincent; Teixeira, Augusto Sharpness of the phase transition for continuum percolation in ${ℝ}^2$ (2016) (https://arxiv.org/abs/1605.05926) | Zbl

[ATT17] Ahlberg, Daniel; Tassion, Vincent; Teixeira, Augusto Existence of an unbounded vacant set for subcritical continuum percolation (2017) (https://arxiv.org/abs/1706.03053) | Zbl

[BDC12] Beffara, Vincent; Duminil-Copin, Hugo The self-dual point of the two-dimensional random-cluster model is critical for $q\ge 1$, Probab. Theory Related Fields, Volume 153 (2012) no. 3-4, pp. 511-542 | DOI | MR | Zbl

[BH57] Broadbent, S. R.; Hammersley, John M. Percolation processes. I. Crystals and mazes, Proc. Cambridge Philos. Soc., Volume 53 (1957), pp. 629-641 | DOI | MR | Zbl

[BR06] Bollobás, Béla; Riordan, Olivier The critical probability for random Voronoi percolation in the plane is 1/2, Probab. Theory Related Fields, Volume 136 (2006) no. 3, pp. 417-468 | DOI | MR | Zbl

[DCGR + 18] Duminil-Copin, Hugo; Goswami, Subhajit; Raoufi, Aran; Severo, Franco; Yadin, Ariel Existence of phase transition for percolation using the Gaussian Free Field (2018) (https://arxiv.org/abs/1806.07733)

[DCGRS19] Duminil-Copin, Hugo; Goswami, Subhajit; Rodriguez, P.-F.; Severo, Franco Equality of critical parameters for GFF level-set percolation (2019) (in preparation)

[DCRT17a] Duminil-Copin, Hugo; Raoufi, Aran; Tassion, Vincent Exponential decay of connection probabilities for subcritical Voronoi percolation in ${ℝ}^d$ (2017) (https://arxiv.org/abs/1705.07978, to appear in Probab. Theory Related Fields) | Zbl

[DCRT17b] Duminil-Copin, Hugo; Raoufi, Aran; Tassion, Vincent Sharp phase transition for the random-cluster and Potts models via decision trees (2017) (https://arxiv.org/abs/1705.03104) | Zbl

[DCT16] Duminil-Copin, Hugo; Tassion, Vincent A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model, Commun. Math. Phys., Volume 343 (2016) no. 2, pp. 725-745 | DOI | MR | Zbl

[Gil61] Gilbert, Edgar N. Random plane networks, J. Soc. Indust. Appl. Math., Volume 9 (1961), pp. 533-543 | DOI | MR

[Gou08] Gouéré, Jean-Baptiste Subcritical regimes in the Poisson Boolean model of continuum percolation, Ann. Probab., Volume 36 (2008) no. 4, pp. 1209-1220 | DOI | MR | Zbl

[GT18] Gouéré, Jean-Baptiste; Théret, Marie Equivalence of some subcritical properties in continuum percolation (2018) (https://arxiv.org/abs/1803.00793) | Zbl

[Hal85] Hall, Peter On continuum percolation, Ann. Probab., Volume 13 (1985) no. 4, pp. 1250-1266 | DOI | MR | Zbl

[LP17] Last, Günter; Penrose, Mathew D.x Lectures on the Poisson process, Institute of Mathematical Statistics Textbooks, 7, Cambridge University Press, 2017 | Zbl

[Men86] Menshikov, Mikhail V. Coincidence of critical points in percolation problems, Dokl. Akad. Nauk SSSR, Volume 288 (1986) no. 6, pp. 1308-1311 | MR

[MR96] Meester, Ronald; Roy, Rahul Continuum percolation, Cambridge University Press, 1996 | Zbl

[MRS94] Meester, Ronald; Roy, Rahul; Sarkar, Anish Nonuniversality and continuity of the critical covered volume fraction in continuum percolation, J. Statist. Phys., Volume 75 (1994) no. 1-2, pp. 123-134 | DOI | MR | Zbl

[OSSS05] O’Donnell, Ryan; Saks, Mickael E.; Schramm, O.; Servedio, Rocco A. Every decision tree has an influential variable, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), IEEE Xplore (2005), pp. 31-39

[Pen17] Penrose, Mathew D. Non-triviality of the vacancy phase transition for the Boolean model (2017) (https://arxiv.org/abs/1706.02197) | Zbl

[Zie16] Ziesche, Sebastian Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on ${ℝ}^d$ (2016) (https://arxiv.org/abs/1607.06211, to appear in Annales de l’Institut Henri Poincaré) | Zbl

[ZS85] Zuev, Sergei A.; Sidorenko, Alexander Continuous models of percolation theory. I, Teor. Mat. Fiz., Volume 62 (1985) no. 2, pp. 51-58 | DOI