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<p>In this paper, we consider a regular tessellation of the Euclidean plane and the sequence of its geometric scalings by negative powers of a fixed integer. We generate iteratively random sets as the union of adjacent tiles from these rescaled tessellations. We encode this geometric construction into a combinatorial object, namely a multitype Galton-Watson tree. Our main result concerns the geometric properties of the limiting planar set. In particular, we show that both box and Hausdorff dimensions coincide and we calculate them in terms of the spectral radius of the reproduction matrix associated with this branching process. We then make that spectral radius explicit in several concrete examples when the regular tessellation is either hexagonal, square or triangular.</p>

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Fractal random sets associated with multitype Galton-Watson trees

  • 期刊论文

  • 发表日期：

    2025年10月5日

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  • Pierre Calka,
  • Yann Demichel

摘要

In this paper, we consider a regular tessellation of the Euclidean plane and the sequence of its geometric scalings by negative powers of a fixed integer. We generate iteratively random sets as the union of adjacent tiles from these rescaled tessellations. We encode this geometric construction into a combinatorial object, namely a multitype Galton-Watson tree. Our main result concerns the geometric properties of the limiting planar set. In particular, we show that both box and Hausdorff dimensions coincide and we calculate them in terms of the spectral radius of the reproduction matrix associated with this branching process. We then make that spectral radius explicit in several concrete examples when the regular tessellation is either hexagonal, square or triangular.