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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 \(r > 1\), \(\theta_r(\lambda)\leq \exp(-c_{\lambda}r)\).
82B43 Percolation
82B26 Phase transitions (general) in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
Full Text: DOI
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