On the capacity functional of the infinite cluster of a Boolean model
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arXiv
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| Natura: | Preprint |
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2016
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| _version_ | 1866929399129964544 |
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| author | Last, Günter Penrose, Mathew D. Zuyev, Sergei |
| author_facet | Last, Günter Penrose, Mathew D. Zuyev, Sergei |
| contents | The original 2017 version of this paper, published in Ann. Appl. Probab., 27, 1678--1801, contains a major gap in the proofs. In the subsequent publication in Ann. Appl. Probab., 34, 3370--3374, 2024, we indicated how to fix this. For convenience of the reader, we here update the original paper to incorporate the suggested fix.
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 $θ_L(t) = P(Z_\infty\cap L \neq \emptyset)$, defined for each compact $L\subset R^d$. We prove for any fixed $L$ and $F_0$ that $θ_L(t)$ 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 $θ_L$ 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 $θ_L(\cdot)$ grows at least linearly at the critical value. This implies that the critical exponent known as $β$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1601_04945 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | On the capacity functional of the infinite cluster of a Boolean model Last, Günter Penrose, Mathew D. Zuyev, Sergei Probability 60K35, 60D05 The original 2017 version of this paper, published in Ann. Appl. Probab., 27, 1678--1801, contains a major gap in the proofs. In the subsequent publication in Ann. Appl. Probab., 34, 3370--3374, 2024, we indicated how to fix this. For convenience of the reader, we here update the original paper to incorporate the suggested fix. 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 $θ_L(t) = P(Z_\infty\cap L \neq \emptyset)$, defined for each compact $L\subset R^d$. We prove for any fixed $L$ and $F_0$ that $θ_L(t)$ 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 $θ_L$ 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 $θ_L(\cdot)$ grows at least linearly at the critical value. This implies that the critical exponent known as $β$ 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. |
| title | On the capacity functional of the infinite cluster of a Boolean model |
| topic | Probability 60K35, 60D05 |
| url | https://arxiv.org/abs/1601.04945 |