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- W2951761353 abstract "Consider a Boolean model in $R^d$ with balls of random, bounded radii with distribution $F_0$, centered at the points of a Poisson process of intensity $t>0$. The capacity functional of the infinite cluster $Z_infty$ is given by $theta_L(t) = BP{ Z_inftycap Lneemptyset}$, defined for each compact $LsubsetR^d$. We prove for any fixed $L$ and $F_0$ that $theta_L(t)$ is infinitely differentiable in $t$, except at the critical value $t_c$; we give a Margulis-Russo type formula for the derivatives. More generally, allowing the distribution $F_0$ to vary and viewing $theta_L$ as a function of the measure $F:=tF_0$, we show that it is infinitely differentiable in all directions with respect to the measure $F$ in the supercritical region of the cone of positive measures on a bounded interval. We also prove that $theta_L(cdot)$ grows at least linearly at the critical value. This implies that the critical exponent known as $beta$ is at most 1 (if it exists) for this model. Along the way, we extend a result of H.~Tanemura (1993), on regularity of the supercritical Boolean model in $d geq 3$ with fixed-radius balls, to the case with bounded random radii." @default.
- W2951761353 created "2019-06-27" @default.
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- W2951761353 date "2016-01-19" @default.
- W2951761353 modified "2023-09-27" @default.
- W2951761353 title "On the capacity functional of the infinite cluster of a Boolean model" @default.
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