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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 1454.82016
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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