On a multiscale continuous percolation model with unbounded defects
The authors study multiscale percolation of unbounded Poisson Boolean models in dimension \(d \geq 2\). The Poisson Boolean model consists of the union of balls with centers following a Poisson process with intensity \(\lambda\) and independent radii following a distribution \(\rho\). The multiscale Poisson Boolean model consists of the union of independent and \(R\)-scaled Poisson Boolean models (level \(i\) having Poisson intensity \(\lambda R^{id}\) and radii distribution \(\rho R^{-i}\)). The main result of the paper is the extension of previously known results for bounded radii distributions to the case of an unbounded distribution \(\rho\). Under tail conditions on \(\rho\) the existence of a critical value \(\lambda_\rho\) is shown: for \(\lambda < \lambda_\rho\) there exists \(R_0\) such that for \(R \geq R_0\) there is no percolation. Another result of the paper is that if the expectation of \(\rho^{2\alpha d}\) is finite, then the expectation of the cluster to the power \(\alpha\) is also finite for small \(\lambda\).
scientific article; zbMATH DE number 2101598