Mean-field bounds for Poisson-Boolean percolation

Abstract

We establish the mean-field bounds $\mathit{\gamma }\ge 1$, $\mathit{\delta }\ge 2$ and $△\ge 2$ on the critical exponents of the Poisson-Boolean continuum percolation model under a moment condition on the radii; these were previously known only in the special case of fixed radii (in the case of γ), or not at all (in the case of δ and △). We deduce these as consequences of the mean-field bound $\mathit{\beta }\le 1$, recently established under the same moment condition [8], using a relative entropy method introduced by the authors in previous work [7].