The fractal cylinder process: existence and connectivity phase transition

On the connectivity of the Poisson process on fractals

For a measure mu supported on a compact connected subset of a Euclidean space which satisfies a uniform d-dimensional decay of the volume of balls we show that the maximal edge in the minimum spanning tree of n indepndent samples from mu is, with high probability (log n/n)^(1/d).

Universal Behavior of Connectivity Properties in Fractal Percolation Models

Partially motivated by the desire to better understand the connectivity phase transition in fractal percolation, we introduce and study a class of continuum fractal percolation models in dimension d greater than or equal to 2. These include a scale invariant version of the classical (Poisson) Boolean model of stochastic geometry and (for d=2) the Brownian loop soup introduced by Lawler and Werner. The models lead to random fractal sets whose connectivity properties depend on a parameter lambda. In this paper we mainly study the transition between a phase where the random fractal sets are total…

双曲空间中胖 Poisson 圆柱的渗流

Percolation of fat Poisson cylinders in hyperbolic space

本文研究双曲空间中“胖”圆柱的 Poisson 过程，通过建立其与半尺度不变随机分形模型的新联系，证明该模型经历渗流相变，并得出 $\mathbb{R}^d$ 中分形球模型具有非空片相的次要结果。

Poisson cylinders in hyperbolic space

We consider the Poisson cylinder model in $d$-dimensional hyperbolic space. We show that in contrast to the Euclidean case, there is a phase transition in the connectivity of the collection of cylinders as the intensity parameter varies. We also show that for any non-trivial intensity, the diameter of the collection of cylinders is infinite.

On fractal faithfulness and fine fractal properties of random variables with independent $\boldsymbol{Q^*}$-digits

We develop a new technique to prove the faithfulness of the Hausdorff--Besicovitch dimension calculation of the family $\varPhi(Q^*)$ of cylinders generated by $Q^*$-expansion of real numbers. All known sufficient conditions for the family $\varPhi(Q^*)$ to be faithful for the Hausdorff--Besicovitch dimension calculation use different restrictions on entries $q_{0k}$ and $q_{(s-1)k}$. We show that these restrictions are of purely technical nature and can be removed. Based on these new results, we study fine fractal properties of random variables with independent $Q^*$-digits.

On new fractal phenomena connected with infinite linear IFS

We establish several new fractal and number theoretical phenomena connected with expansions which are generated by infinite linear iterated function systems. First of all we show that the systems $Φ$ of cylinders of generalized Lüroth expansions are, generally speaking, not faithful for the Hausdorff dimension calculation. Using Yuval Peres' approach, we prove sufficient conditions for the non-faithfulness of such families of cylinders. On the other hand, rather general sufficient conditions for the faithfulness of such covering systems are also found. As a corollary of our main results, we ob…