Related papers: Fractal Percolations
In this paper we analyse random walk on a fractal structure, specifi- cally fractal curves, using the recently develped calculus for fractal curves. We consider only unbiased random walk on the fractal stucture and find out the…
Jigsaw percolation is a nonlocal process that iteratively merges connected clusters in a deterministic "puzzle graph" by using connectivity properties of a random "people graph" on the same set of vertices. We presume the Erdos--Renyi…
We describe infinite clusters which arise in nearest-neighbour percolation for so-called cocycle measures on the square lattice. These measures arise naturally in the study of random transformations. We show that infinite clusters have a…
We study dimensional properties of visible parts of fractal percolation in the plane. Provided that the dimension of the fractal percolation is at least 1, we show that, conditioned on non-extinction, almost surely all visible parts from…
We consider a fractal with a variable fractal dimension, which is a generalization of the well known triadic Cantor set. In contrast with the usual Cantor set, the fractal dimension is controlled using a scaling factor, and can vary from…
We analyze the fragmentation behavior of random clusters on the lattice under a process where bonds between neighboring sites are successively broken. Modeling such structures by configurations of a generalized Potts or random-cluster model…
We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realization of a fractal…
This is the first of a series of three articles that treats fracture localization as a critical phenomenon. This first article establishes a statistical mechanics based on ensemble averages when fluctuations through time play no role in…
We give a necessary and sufficient condition to achieve the most uncertain exponent in the fractal uncertainty principle of discrete Cantor sets. The condition will be described as distributed spectral pairs, which is a generalization of…
This article is devoted to sets having the Moran structure. The main attention is given to topological, metric, and fractal properties of certain sets whose elements have restrictions on using digits or combinations of digits in own…
A solid wooden cube fragments into pieces as we sequentially drill holes through it randomly. This seemingly straightforward observation encompasses deep and nontrivial geometrical and probabilistic behavior that is discussed here.…
The advantages of introducing a fractal viewpoint in the field of combustion is emphasized. It is shown that the condition for perfect combustion of a collection of drops is the self-similarity of the distribution.
We analyze damage nucleation and localization in the random fuse model with strong disorder using numerical simulations. In the initial stages of the fracture process, damage evolves in an uncorrelated manner, resembling percolation.…
The fractal property stipulates that the same physical laws apply for different scales of a given physics. This property is applied to particles and nuclei, in order to study the possibility to use it to help for determination of unknown…
The theory of resource distribution in self-organizing systems on the basis of the fractal-cluster method has been presented. In turn, the fractal-cluster method is based on the fractal-cluster relations of V.P. Burdakov and the analytical…
Probabilistic circuits (PCs) represent a probability distribution as a computational graph. Enforcing structural properties on these graphs guarantees that several inference scenarios become tractable. Among these properties, structured…
The development of percolation theory was historically shaped by its numerous applications in various branches of science, in particular in statistical physics, and was mainly constrained to the case of Euclidean spaces. One of its central…
Selfsimilar space-time fractal fluctuations are generic to dynamical systems in nature such as atmospheric flows, heartbeat patterns, population dynamics, etc. The physics of the long-range correlations intrinsic to fractal fluctuations is…
We consider the fragmentation process with mass loss and discuss self-similar properties of the arising structure both in time and space focusing on dimensional analysis. This exhibits a spectrum of mass exponents $\theta$, whose exact…
We study diffusion processes in anomalous spacetimes regarded as models of quantum geometry. Several types of diffusion equation and their solutions are presented and the associated stochastic processes are identified. These results are…