Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Almost sure convergence and second moments of geometric functionals of fractal percolation

Michael A. Klatt, Steffen Winter

TL;DR

The paper establishes almost sure convergence for the rescaled intrinsic volumes $Z_n^k=M^{(k-D)n}V_k(F_n)$ of fractal percolation, showing $Z_n^k\to Z_\infty^k=\overline{\mathcal{V}}_k(F)\,W_\infty$ with a clear factorization between deterministic geometric content and a stochastic Galton–Watson limit $W_\infty$. By leveraging a renewal theorem for branching random walks, it also derives the complete covariance structure among the limit variables and computes exact rates for the convergence of the first two moments (volume and surface area) of the finite approximations. The results extend prior work by moving from expectations to almost sure limits and second moments, and by providing explicit variance-covariance formulas. This deepens the understanding of the interplay between the random offspring structure and the deterministic self-similar geometry in fractal percolation, with potential connections to percolation thresholds via moment-based indicators.

Abstract

We determine almost sure limits of rescaled intrinsic volumes of the construction steps of fractal percolation in $\mathbb{R}^d$ for any dimension $d\geq 1$. We observe a factorization of these limit variables which allows, in particular, to determine their expectations and covariance structure. We also show convergence of rescaled expectations and variances of the intrinsic volumes of the construction steps to expectations and variances of the limit variables and give rates for this convergence in some cases. These results significantly extend our previous work that addressed only limits of expectations of intrinsic volumes.

Almost sure convergence and second moments of geometric functionals of fractal percolation

TL;DR

The paper establishes almost sure convergence for the rescaled intrinsic volumes of fractal percolation, showing with a clear factorization between deterministic geometric content and a stochastic Galton–Watson limit . By leveraging a renewal theorem for branching random walks, it also derives the complete covariance structure among the limit variables and computes exact rates for the convergence of the first two moments (volume and surface area) of the finite approximations. The results extend prior work by moving from expectations to almost sure limits and second moments, and by providing explicit variance-covariance formulas. This deepens the understanding of the interplay between the random offspring structure and the deterministic self-similar geometry in fractal percolation, with potential connections to percolation thresholds via moment-based indicators.

Abstract

We determine almost sure limits of rescaled intrinsic volumes of the construction steps of fractal percolation in for any dimension . We observe a factorization of these limit variables which allows, in particular, to determine their expectations and covariance structure. We also show convergence of rescaled expectations and variances of the intrinsic volumes of the construction steps to expectations and variances of the limit variables and give rates for this convergence in some cases. These results significantly extend our previous work that addressed only limits of expectations of intrinsic volumes.
Paper Structure (9 sections, 12 theorems, 56 equations, 2 figures)

This paper contains 9 sections, 12 theorems, 56 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Almost sure convergence of rescaled intrinsic volumes
  3. Additional results: Analysis of the finite approximations
  4. Random iterated function systems and a renewal theorem for branching random walks
  5. Fractal percolation as a random self-similar set
  6. A branching random walk associated with $F$.
  7. Proof of Theorem \ref{['thm:as-conv-Vk']}
  8. Variance of the volume of $F_n$
  9. Expectation and variance of the surface area of $F_n$

Key Result

Theorem 2.1

KW18 Let $F$ be a fractal percolation on $[0,1]^d$ with parameters $M\in{\mathbb N}_{\geq 2}$ and $p\in(0,1]$. Let $D$ be as in eq:dimF. Then, for each $k\in\{0,\ldots,d\}$, the limit exists and is given by the expression

Figures (2)

  • Figure 1: Semi-logarithmic plots of the variances and the absolute values of the covariances of $Z^k_{\infty}$ in ${\mathbb R}^2$ as functions of $p$ for different values of $M\leq 20$ (indicated by the color scale), see \ref{['eq:var']} and \ref{['eq:cov']}. For $M\to\infty$, all variances and covariances converge to zero.
  • Figure 2: Semi-logarithmic plots of the variance of $Z^0_{\infty}$ rescaled by $M^2$ as functions of $p$ for $M=2$ (left), 3 (center), and 1000 (right). The shaded areas for $M=2$ and 3 indicate rigorously known bounds on the percolation threshold Don15 as in KW18. The vertical line for $M=1000$ indicates an empirical estimate of $p_{c,NN}$ provided by simulations NZ00.

Theorems & Definitions (15)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Corollary 3.1
  • Theorem 3.2
  • Corollary 3.2
  • Theorem 4.1
  • ...and 5 more