Geometric functionals of fractal percolation

TL;DR

The paper introduces geometric functionals for fractal percolation by extending intrinsic volumes to the fractal limit via rescaled limits of finite approximations. It proves the existence of these limit functionals and derives explicit formulas in dimensions 1 and 2, including for the closed complements, with rapid exponential convergence from finite steps. While zeros of these functionals correlate with percolation behavior, the connections to percolation thresholds are nuanced: Euler-characteristic based criteria yield bounds that are informative but not tight, and percolation can occur on lower-dimensional subsets, limiting their predictive power for $p_c$. The results provide concrete, computable descriptors of the random fractal geometry and offer potential applications as robust geometric shape descriptors for random fractals beyond plain fractal dimension.

Abstract

Fractal percolation exhibits a dramatic topological phase transition, changing abruptly from a dust-like set to a system spanning cluster. The transition points are unknown and difficult to estimate. In many classical percolation models the percolation thresholds have been approximated well using additive geometric functionals, known as intrinsic volumes. Motivated by the question whether a similar approach is possible for fractal models, we introduce corresponding geometric functionals for the fractal percolation process $F$. They arise as limits of expected functionals of finite approximations of $F$. We establish the existence of these limit functionals and obtain explicit formulas for them as well as for their finite approximations.

Figures (8)

• Figure 1: Finite approximations of fractal percolation: realizations for different values of the survival probability $p$ and the linear number of subdivisions $M$. 'Percolating' clusters that span the system in vertical and horizontal direction are highlighted.
• Figure 2: Illustration of the three intrinsic volumes in ${\mathbb R}^2$ -- area, boundary length and Euler characteristic (i.e., #components $-$ #holes).
• Figure 3: Illustration of the sets $F_n^j$ as subsets of $F_n$ (for $M=2$ and $n=6$) and of an intersection $\bigcap_{j\in T} F_n^j$. The number of subsets $F_n^j$ is by definition always $M^2$.
• Figure 4: Rescaled expected Euler characteristic of finite approximations $F_n$ (left) and their closed complements $C_n$ (right) as functions of the survival probability $p$ for $M=2$ (top), $M=3$ (center) and $M=4$ (bottom). Each plot compares finite approximations with increasing $n$ to the limit curve ($n=\infty$), that is, to $p\mapsto \overline{\mathcal{V}}_0(F(p))$ given in Theorem \ref{['thm:Vk-limit-dim2']} (left) and $p\mapsto -\overline{\mathcal{V}}_0^c(F(p))$ given in Theorem \ref{['thm:Vck-limit-dim2']}. The shaded areas indicate the rigorously known bounds on the percolation threshold, see \ref{['eq:pc-bounds2']}.
• Figure 5: For increasing values of the number of subdivisions $M$ (color coded), the rescaled expected Euler characteristic of fractal percolation (left) and its complement (right) are plotted as functions of $p$. The limiting curve (red) for $M\to\infty$ corresponds to the mean Euler characteristic per site (rescaled by the intensity) of site percolation on ${\mathbb Z}^2$ with eight or four neighbors, respectively.

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 5.1
• proof : Proof of Theorem \ref{['thm:Vk-limit-general']}
• Proposition 5.2
• proof
• Corollary 5.3
• proof
• Proposition 5.4