Mean-field bounds for Poisson-Boolean percolation. (English) Zbl 1524.60264

Summary: We establish the mean-field bounds \(\gamma \ge 1\), \(\delta \ge 2\) and \(\Delta \ge 2\) 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 \(\gamma \)), or not at all (in the case of \(\delta\) and \(\Delta \)). We deduce these as consequences of the mean-field bound \(\beta \le 1\), recently established under the same moment condition [H. Duminil-Copin et al., Ann. Henri Lebesgue 3, 677–700 (2020; Zbl 1454.82016)], using a relative entropy method introduced by the authors in previous work [“Upper bounds on the one-arm exponent for dependent percolation models”, Probab. Theory Relat. Fields 185, No. 1–2, 41–88 (2023; doi:10.1007/s00440-022-01176-3)].


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
60D05 Geometric probability and stochastic geometry


Zbl 1454.82016
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