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Subcritical phase of $$d$$-dimensional Poisson-Boolean percolation and its vacant set. (Phase sous-critique du modèle de percolation Poisson-Booléen et de son complémentaire en dimension $$d$$.) (English. French summary) Zbl 07249463
For every $$r>0$$, define the two functions of $$\lambda$$ as $$\theta_r(\lambda):=\mathbf{P}_{\lambda}[0\leftrightarrow \partial B_r]$$ and $$\theta(\lambda):=\lim_{r\rightarrow \infty}\theta_r(\lambda).$$ Define the critical parameter $$\lambda_c =\lambda_c(d)$$ of the model by the formula $$\lambda_c := \inf\{\lambda\geq 0 : \theta(\lambda) > 0\}$$. The authors introduce the another critical parameter to discuss Poisson-Boolean percolation as $$\widetilde{\lambda}_c:=\inf \left\{\lambda \geq 0:\underset{r>0}{\inf }\mathbb{P}_{\lambda }[B_{\lambda}\leftrightarrow \partial B_{2r}]>0\right\}$$. The authors prove the following main result:
Theorem 1.2 (Sharpness for Poisson-Boolean percolation). – Fix $$d\geq 2$$ and assume that $$\underset{\mathbb{R}_+}{\int }r^{5d-3}d\mu (r)<\infty.$$ Then, we have that $$\lambda_c = \widetilde{\lambda}_c$$. Furthermore, there exists $$c > 0$$ such that $$\theta(\lambda)> c(\lambda-\lambda_c)$$ for any $$\lambda\geq \lambda_c$$.
The authors give a brief description of the general strategy to prove the main theorem. Three properties of the Poisson-Boolean percolation are introduced. Then, the authors present some new results concerning the behavior of Poisson-Boolean percolation when $$\lambda<\widetilde{\lambda}_c$$. If there exists $$c > 0$$ such that $$\mu[r,\infty]\leq \exp(-cr)$$ for every $$r\geq 1$$, then, for every $$\lambda<\widetilde{\lambda}_c$$, the authors prove that there exists $$c_{\lambda}> 0$$ such that for every $$r > 1$$, $$\theta_r(\lambda)\leq \exp(-c_{\lambda}r)$$.
##### MSC:
 82B43 Percolation 82B26 Phase transitions (general) in equilibrium statistical mechanics 82B27 Critical phenomena in equilibrium statistical mechanics
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##### References:
 [1] Aizenman, Michael; Barsky, David J., Sharpness of the phase transition in percolation models, Commun. Math. Phys., 108, 3, 489-526 (1987) · Zbl 0618.60098 [2] Aizenman, Michael; Barsky, David J.; Fernández, Roberto, The phase transition in a general class of Ising-type models is sharp, J. Statist. Phys., 47, 3-4, 343-374 (1987) [3] Ahlberg, Daniel; Tassion, Vincent; Teixeira, Augusto, Sharpness of the phase transition for continuum percolation in $$\mathbb{R}^2 (2016)$$ · Zbl 1404.60143 [4] Ahlberg, Daniel; Tassion, Vincent; Teixeira, Augusto, Existence of an unbounded vacant set for subcritical continuum percolation (2017) · Zbl 1401.60173 [5] 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, 153, 3-4, 511-542 (2012) · Zbl 1257.82014 [6] Broadbent, S. R.; Hammersley, John M., Percolation processes. I. Crystals and mazes, Proc. Cambridge Philos. Soc., 53, 629-641 (1957) · Zbl 0091.13901 [7] Bollobás, Béla; Riordan, Olivier, The critical probability for random Voronoi percolation in the plane is 1/2, Probab. Theory Related Fields, 136, 3, 417-468 (2006) · Zbl 1100.60054 [8] Duminil-Copin, Hugo; Goswami, Subhajit; Raoufi, Aran; Severo, Franco; Yadin, Ariel, Existence of phase transition for percolation using the Gaussian Free Field (2018) [9] Duminil-Copin, Hugo; Goswami, Subhajit; Rodriguez, P.-F.; Severo, Franco, Equality of critical parameters for GFF level-set percolation (2019) [10] Duminil-Copin, Hugo; Raoufi, Aran; Tassion, Vincent, Exponential decay of connection probabilities for subcritical Voronoi percolation in $$\mathbb{R}^d (2017)$$ · Zbl 07030876 [11] Duminil-Copin, Hugo; Raoufi, Aran; Tassion, Vincent, Sharp phase transition for the random-cluster and Potts models via decision trees (2017) · Zbl 07003145 [12] 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., 343, 2, 725-745 (2016) · Zbl 1342.82026 [13] Gilbert, Edgar N., Random plane networks, J. Soc. Indust. Appl. Math., 9, 533-543 (1961) · Zbl 0112.09403 [14] Gouéré, Jean-Baptiste, Subcritical regimes in the Poisson Boolean model of continuum percolation, Ann. Probab., 36, 4, 1209-1220 (2008) · Zbl 1148.60077 [15] Gouéré, Jean-Baptiste; Théret, Marie, Equivalence of some subcritical properties in continuum percolation (2018) · Zbl 1428.62425 [16] Hall, Peter, On continuum percolation, Ann. Probab., 13, 4, 1250-1266 (1985) · Zbl 0588.60096 [17] Last, Günter; Penrose, Mathew D.x, Lectures on the Poisson process, 7 (2017), Cambridge University Press · Zbl 1392.60004 [18] Menshikov, Mikhail V., Coincidence of critical points in percolation problems, Dokl. Akad. Nauk SSSR, 288, 6, 1308-1311 (1986) [19] Meester, Ronald; Roy, Rahul, Continuum percolation (1996), Cambridge University Press · Zbl 0858.60092 [20] Meester, Ronald; Roy, Rahul; Sarkar, Anish, Nonuniversality and continuity of the critical covered volume fraction in continuum percolation, J. Statist. Phys., 75, 1-2, 123-134 (1994) · Zbl 0828.60083 [21] 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), 31-39 (2005), IEEE Xplore [22] Penrose, Mathew D., Non-triviality of the vacancy phase transition for the Boolean model (2017) · Zbl 1394.60101 [23] Ziesche, Sebastian, Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on $$\mathbb{R}^d (2016)$$ · Zbl 1391.60246 [24] Zuev, Sergei A.; Sidorenko, Alexander, Continuous models of percolation theory. I, Teor. Mat. Fiz., 62, 2, 51-58 (1985)
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