论文标题
泊松布尔渗透几乎清晰的清晰度
Almost sharp sharpness for Poisson Boolean percolation
论文作者
论文摘要
我们考虑在$ \ mathbb r^d $上进行的泊松布尔渗透，而半径上有幂律分布，有限的$ d $ - momment售价$ d \ ge 2 $。我们证明，除了可数量的幂律分布外，所有人都会出现亚临界的清晰度。这扩展了Duminil-copin--raoufi-tassion的结果，在这种情况下，在半临界的假设是半径分布具有5D-3 $有限矩的假设之下。我们的证明技术与他们的论文不同：我们不使用随机算法，而依靠布尔渗透的特定独立性，该特性是从基础泊松过程继承的。 对于任何有限的$ d $ amoment的分布，我们还证明了超临界的清晰度，以及当截断进入无穷大时，截断分布的关键参数的连续性。
We consider Poisson Boolean percolation on $\mathbb R^d$ with power-law distribution on the radius with a finite $d$-moment for $d\ge 2$. We prove that subcritical sharpness occurs for all but a countable number of power-law distributions. This extends the results of Duminil-Copin--Raoufi--Tassion where subcritical sharpness is proved under the assumption that the radii distribution has a $5d-3$ finite moment. Our proofs techniques are different from their paper: we do not use randomized algorithm and rely on specific independence properties of Boolean percolation, inherited from the underlying Poisson process. We also prove supercritical sharpness for any distribution with a finite $d$-moment and the continuity of the critical parameter for the truncated distribution when the truncation goes to infinity.
