论文标题
分形圆柱体过程：存在和连通性相变
The fractal cylinder process: existence and connectivity phase transition
论文作者
论文摘要
我们考虑了泊松缸模型的半规模不变版本，该版本以自然方式诱导随机分形集。我们表明，每当$ d \ geq 4. $我们确定存在相变的临界点的确切值时，该随机分形表现出对任何维度$ d \ geq 2，$和连接相转换的存在相变的存在，我们表明分形集合在此关键点几乎肯定是空的。 分析连接相变的关键成分是考虑将整个过程限制到子空间上。我们表明，这种限制会导致分形椭圆形模型，我们将详细描述，因为这是获得我们的主要结果的关键。 此外，我们还确定了分形组的几乎确定的Hausdorff尺寸。
We consider a semi-scale invariant version of the Poisson cylinder model which in a natural way induces a random fractal set. We show that this random fractal exhibits an existence phase transition for any dimension $d\geq 2,$ and a connectivity phase transition whenever $d\geq 4.$ We determine the exact value of the critical point of the existence phase transition, and we show that the fractal set is almost surely empty at this critical point. A key ingredient when analysing the connectivity phase transition is to consider a restriction of the full process onto a subspace. We show that this restriction results in a fractal ellipsoid model which we describe in detail, as it is key to obtaining our main results. In addition we also determine the almost sure Hausdorff dimension of the fractal set.
