Patterns in random fractals

Journal of Statistical Planning and Inference, 2003

We determine the fractal dimension d f of inÿnite spherically symmetric random trees (all vertices at distance n from the root have the same degree dn where {dn} are independent random variables). If dn takes the value 3 or 2 with probabilities qn and 1−qn, then d f =(log 2) limn nqn+1 a.s. We show how d f is closely related to the type of the simple random walks (SRW) on trees. We prove that the SRW is a.s. transient if d f ¿ 2 a.s. and a.s. recurrent if d f ¡ 2 a.s. and if d f = 2 a.s. we obtain a.s. transience or recurrence. We also consider another type of random trees which are corresponding to branching processes in varying environments. In particular, we consider a tree such that the degrees of the vertices at distance n from the root are independent identically distributed (iid) random variables following the distribution of a random variable that takes the value 3 or 2 with probabilities qn and 1−qn respectively. These iid random variables are also independent of the degrees of the vertices of the other generations. We prove that the SRW is a.s. recurrent if and only if d f 6 2 a.s. We also prove for such trees that d f = limn nqn + 1 a.s.

Electronic Journal of Probability, 2010

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 ≥ 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 λ. In this paper we mainly study the transition between a phase where the random fractal sets are totally disconnected and a phase where they contain connected components larger than one point. In particular, we show that there are connected components larger than one point at the unique value of λ that separates the two phases (called the critical point). We prove that such a behavior occurs also in Mandelbrot's fractal percolation in all dimensions d ≥ 2. Our results show that it is a generic feature, independent of the dimension or the precise definition of the model, and is essentially a consequence of scale invariance alone. Furthermore, for d = 2 we prove that the presence of connected components larger than one point implies the presence of a unique, unbounded, connected component.

Latin American journal of probability and mathematical statistics

We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N^d equal subcubes, k of which are retained while the others are discarded. The procedure is then iterated inside the retained cubes at all smaller scales. We show that the (properly rescaled) percolation critical value of this model converges to the critical value of ordinary site percolation on a particular d-dimensional lattice as N tends to infinity. This is analogous to the result of Falconer and Grimmett that the critical value for Mandelbrot fractal percolation converges to the critical value of site percolation on the same d-dimensional lattice. In the fat fractal percolation model, subcubes are retained with probability p_n at step n of the construction, where (p_n) is a non-decreasing sequence with \prod p_n > 0. The Lebesgue measure of the limit set is positive a.s. given non-extinction. We prove that either the set of co...

arXiv (Cornell University), 2019

We show that there exists 0 < α 0 < 1 (depending on the parameters) such that the fractal percolation is almost surely purely α-unrectifiable for all α > α 0. Contents 1. Introduction 1 2. Idea of the proof 3 3. Fractal percolation model 4 4. Pure 1-unrectifiability 5 5. Existence of hereditarily good cubes 14 6. Pure α-unrectifiability 19 7. Appendix A: Definition of zoom levels 24 8. Appendix B: Broken line approximations 30 9. Appendix C: Increase of length 38 Acknowledgement 46 References 46

Forum Mathematicum, 2000

V-variable fractals, for V = 1, 2, 3,. .. , interpolate between random homogeneous fractals and random recursive fractals. We compute the almost sure Hausdorff dimension of V-variable fractals satisfying the uniform open set condition. Important roles are played by the notion of a neck, leading to spatial homogeneity at various levels of magnification, and a variant of the Furstenberg Kesten theorem for products of certain random V × V matrices.

Probability in the Engineering and Informational Sciences, 2007

In this article we determine a formula for fractal and resistance dimensions of two models of uniformly bounded random trees. The type (transient or recurrent) of the random walk on such trees is ascribed, to some extent, to these dimensions. The results presented in this article generalize some of the results of [6] and [7].

Journal of Statistical Physics, 1984

We discuss the fractal dimension of the infinite cluster at the percolation threshold. Using sealing theory and renormalization group we present an explicit expression for the two-point correlation function within percolation clusters. The fractaI dimension is given by direct integration of this function.

Ergodic Theory and Dynamical Systems, 2013

We study the dimension of code tree fractals, a class of fractals generated by a set of iterated function systems. We first consider deterministic affine code tree fractals, extending to the code tree fractal setting the classical result of Falconer and Solomyak on the Hausdorff dimension of self-affine fractals generated by a single iterated function system. We then calculate the almost sure Hausdorff, packing and box counting dimensions of a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous Markov constructions.

Electronic Journal of Probability, 2008

We study Mandelbrot's percolation process in dimension d ≥ 2. The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube [0, 1] d in N d subcubes, and independently retaining or discarding each subcube with probability p or 1 − p respectively. This step is then repeated within the retained subcubes at all scales. As p is varied, there is a percolation phase transition in terms of paths for all d ≥ 2, and in terms of (d − 1)-dimensional "sheets" for all d ≥ 3. For any d ≥ 2, we consider the random fractal set produced at the path-percolation critical value p c (N, d), and show that the probability that it contains a path connecting two opposite faces of the cube [0, 1] d tends to one as N → ∞. As an immediate consequence, we obtain that the above probability has a discontinuity, as a function of p, at p c (N, d) for all N sufficiently large. This had previously been proved only for d = 2 (for any N ≥ 2). For d ≥ 3, we prove analogous results for sheet-percolation. In dimension two, Chayes and Chayes proved that p c (N, 2) converges, as N → ∞, to the critical density p c of site percolation on the square lattice. Assuming the existence of

Fractals in Physics, 1986

The fractal dimensions of various types of Jntersection sets of random fractals are discussed. ffris frÀipi a better understanding oÍ'issues concernlng-upper critical dimensionality' F'lory upp"oiimátions or universality oi critical behaviour w'lth respect to different excluded volume màttranisms. Diffusion on sel?-intersecting fractals, and on fractals with local bridges' is also iónjiaereA in the light of the above discussion, and accurate results for the relative exponents' obtained from ana'lysis of Monte-Carlo generated enumeratlons, are presented.