Fat fractal percolation and k-fractal percolation

Erik I. Broman, Tim van de Brug, Federico Camia, Matthijs Joosten, Ronald Meester

    Research output: Contribution to journalArticle

    Abstract

    We consider two variations on theMandelbrot fractal percolationmodel. In the k-fractal percolation model, the d-dimensional unit cube is divided in Nd 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→∞ This is analogous to the result of Falconer and Grimmett (1992) 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 pn at step n of the construction, where (pn)n≥ is a non-decreasing sequence with π n=1 pn > 0. The Lebesgue measure of the limit set is positive a.s. given nonextinction. We prove that either the set of connected components larger than one point has Lebesgue measure zero a.s. or its complement in the limit set has Lebesgue measure zero a.s.

    Original languageEnglish (US)
    Pages (from-to)279-301
    Number of pages23
    JournalAlea
    Volume9
    Issue number2
    StatePublished - Dec 1 2012

    Fingerprint

    Fractal
    Critical value
    Lebesgue Measure
    Limit Set
    Converge
    Unit cube
    Zero
    Connected Components
    Regular hexahedron
    Complement
    Model

    Keywords

    • Critical value.
    • Crossing probability
    • Fractal percolation
    • Random fractals

    ASJC Scopus subject areas

    • Statistics and Probability

    Cite this

    Broman, E. I., van de Brug, T., Camia, F., Joosten, M., & Meester, R. (2012). Fat fractal percolation and k-fractal percolation. Alea, 9(2), 279-301.

    Fat fractal percolation and k-fractal percolation. / Broman, Erik I.; van de Brug, Tim; Camia, Federico; Joosten, Matthijs; Meester, Ronald.

    In: Alea, Vol. 9, No. 2, 01.12.2012, p. 279-301.

    Research output: Contribution to journalArticle

    Broman, EI, van de Brug, T, Camia, F, Joosten, M & Meester, R 2012, 'Fat fractal percolation and k-fractal percolation', Alea, vol. 9, no. 2, pp. 279-301.
    Broman EI, van de Brug T, Camia F, Joosten M, Meester R. Fat fractal percolation and k-fractal percolation. Alea. 2012 Dec 1;9(2):279-301.
    Broman, Erik I. ; van de Brug, Tim ; Camia, Federico ; Joosten, Matthijs ; Meester, Ronald. / Fat fractal percolation and k-fractal percolation. In: Alea. 2012 ; Vol. 9, No. 2. pp. 279-301.
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