On random fractals with infinite branching: definition, measurability, dimensions

The calculation of multi-fractal properties of directed random walks on hierarchic trees with continuous branching

We consider the hierarchic tree Random Energy Model with continuous branching and calculate the moments of the corresponding partition function. We establish the multifractal properties of those moments. We derive formulas for the normal distribution of random variables, as well as for the general case. We compare our results for the moments of partition function with corresponding results of logarithmic 1-d REM and conjecture a specific power-law tail for the partition function distribution in the high-temperature phase. Our results establish a connection between reaction-diffusion equations …

Central limit theorems for the spectra of classes of random fractals

We discuss the spectral asymptotics of some open subsets of the real line with random fractal boundary and of a random fractal, the continuum random tree. In the case of open subsets with random fractal boundary we establish the existence of the second order term in the asymptotics almost surely and then determine when there will be a central limit theorem which captures the fluctuations around this limit. We will show examples from a class of random fractals generated from Dirichlet distributions as this is a relatively simple setting in which there are sets where there will and will not be a…

Probabilistic and fractal aspects of Levy trees

We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete Galton-Watson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted R-trees, which is equipped with the Gromov-Hausdorff distance. To construct Lévy trees, we make use of the coding by the height process which was studied in detail in previous work. We then investigate various probabilistic properties of Lévy trees. In particular we establish a branching property analogous to the w…

分形渗流与随机集的Minkowski和

Minkowski sum of fractal percolation and random sets

本文证明分形渗流Minkowski和的命中概率可由容量刻画，并推广至一般随机集，如随机游走和临界分支随机游走的轨迹。

与多类型 Galton-Watson 树相关的分形随机集

Fractal random sets associated with multitype Galton-Watson trees

该研究用多类型 Galton-Watson 树编码由正则镶嵌相邻瓦片迭代生成的随机平面集，证明极限集的盒维数与 Hausdorff 维数相等，并由再生矩阵谱半径计算，还给出三类镶嵌示例。

Euclidean Embedding of the Poisson Weighted Infinite Tree and Application to Mobility Models

Continuous time branching models are used to create random fractals in a Euclidean space, whose Hausdorff dimension is controlled by an input parameter. Finite realizations are applied in modelling the set of sites visited in models of human and animal mobility.