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Subcritical regimes in the Poisson Boolean percolation on Ahlfors regular spaces

Yutaka Takeuchi

Abstract

The Poisson Boolean percolation on a metric measure space is one of the percolation models. Intuitively, this model is obtained by collecting random balls whose centers form a Poisson point process. In 2008, Gouéré proved that for $n \geq 2$, the Poisson Boolean percolation on $\mathbb{R}^n$ has the subcritical regime if and only if the radius distribution has finite $n$-th moment. In this paper, we extend Gouéré's result to Ahlfors regular metric measure spaces.

Subcritical regimes in the Poisson Boolean percolation on Ahlfors regular spaces

Abstract

The Poisson Boolean percolation on a metric measure space is one of the percolation models. Intuitively, this model is obtained by collecting random balls whose centers form a Poisson point process. In 2008, Gouéré proved that for , the Poisson Boolean percolation on has the subcritical regime if and only if the radius distribution has finite -th moment. In this paper, we extend Gouéré's result to Ahlfors regular metric measure spaces.

Paper Structure

This paper contains 11 sections, 18 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Motivation and historical backgrounds
  3. Settings and main results
  4. Existence of subcritical regimes
  5. Nonexistence of the subcritical behavior
  6. Applications
  7. Poisson Boolean percolation on Riemannian manifold with nonnegative Ricci curvature
  8. Ultrametric space case
  9. Unbounded Sierpinski gasket
  10. Inhomogeneous continuum percolation on the Euclidean space
  11. Snowflake of an Ahlfors regular space

Key Result

Lemma 1.2

Let $(S, d, \mu)$ be an $s$-Ahlfors regular space. Let $\sigma > C_\textup{V}^\frac{2}{C_\textup{V}}$. Then $(S, d)$ is $\sigma$-uniformly perfect, that is, $B(x, \sigma r) \setminus B(x, r)$ is nonempty for all $x \in S$ and $r > 0$.

Theorems & Definitions (32)

  • Definition 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 22 more