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Chemical distance in the Poisson Boolean model with regularly varying diameters

Peter Gracar, Marilyn Korfhage

TL;DR

The paper analyzes the chemical distance in a Poisson-Boolean base model with regularly varying diameters and rotation-invariant convex grains. In the robust but non-dense regime, it proves ultrasmall scaling for the distance between distant points: $\operatorname{dist}(\mathbf{x},\mathbf{y})$ is of order $\log\log|x-y|$ with an explicit constant determined by the tail indices through $\kappa=\underset{s\in M}{\operatorname{argmax}} \frac{\min\{d-s,s\}}{\alpha_s-s}$. The main novelty is the explicit, parameter-dependent constant and the dichotomy based on the tail indices, alongside a rigorous geometric reduction to boxes and a two-stage sprinkling argument to bound both ends of typical paths. The methodology combines geometric lemmas (reducing convex grains to boxes), Palm-type conditioning, and careful probabilistic constructions (A_n^x, threshold sequences) to obtain matching lower and upper bounds. The results are illustrated through diverse examples, including ellipsoids with varying axis-tail behaviors and random triangular grains, highlighting the broad applicability of the ultrasmall scaling phenomenon in spatial random graphs without needing long-range edges.

Abstract

We study the Poisson Boolean model with convex bodies which are rotation-invariant distributed. We assume that the convex bodies have regularly varying diameters with indices $-α_1\geq \dots\geq-α_d$ where $α_k >0$ for all $k\in\{1,\dots,d\}.$ It is known that a sufficient condition for the robustness of the model, i.e. the union of the convex bodies has an unbounded connected component no matter what the intensity of the underlying Poisson process is, is that there exists some $k\in\{1,\dots,d\}$ such that $α_k<\min\{2k,d\}$. To avoid that this connected component covers all of $\mathbb{R}^d$ almost surely we also require $α_k> k$ for all $k\in\{1,\dots,d\}$. We show that under these assumptions, the chemical distance of two far apart vertices $\mathbf{x}$ and $\mathbf{y}$ behaves like $c\log\log|x-y|$ as $|x-y|\rightarrow \infty$, with an explicit and very surprising constant $c$ that depends only on the model parameters. We furthermore show that if there exists $k$ such that $α_k\leq k$, the chemical distance is smaller than $c\log\log|x-y|$ for all $c>0$ and that if $α_k\geq\min\{2k,d\}$ for all $k$, it is bigger than $c\log\log|x-y|$ for all $c>0$.

Chemical distance in the Poisson Boolean model with regularly varying diameters

TL;DR

The paper analyzes the chemical distance in a Poisson-Boolean base model with regularly varying diameters and rotation-invariant convex grains. In the robust but non-dense regime, it proves ultrasmall scaling for the distance between distant points: is of order with an explicit constant determined by the tail indices through . The main novelty is the explicit, parameter-dependent constant and the dichotomy based on the tail indices, alongside a rigorous geometric reduction to boxes and a two-stage sprinkling argument to bound both ends of typical paths. The methodology combines geometric lemmas (reducing convex grains to boxes), Palm-type conditioning, and careful probabilistic constructions (A_n^x, threshold sequences) to obtain matching lower and upper bounds. The results are illustrated through diverse examples, including ellipsoids with varying axis-tail behaviors and random triangular grains, highlighting the broad applicability of the ultrasmall scaling phenomenon in spatial random graphs without needing long-range edges.

Abstract

We study the Poisson Boolean model with convex bodies which are rotation-invariant distributed. We assume that the convex bodies have regularly varying diameters with indices where for all It is known that a sufficient condition for the robustness of the model, i.e. the union of the convex bodies has an unbounded connected component no matter what the intensity of the underlying Poisson process is, is that there exists some such that . To avoid that this connected component covers all of almost surely we also require for all . We show that under these assumptions, the chemical distance of two far apart vertices and behaves like as , with an explicit and very surprising constant that depends only on the model parameters. We furthermore show that if there exists such that , the chemical distance is smaller than for all and that if for all , it is bigger than for all .

Paper Structure

This paper contains 10 sections, 6 theorems, 100 equations, 7 figures.

Table of Contents

  1. Introduction and result
  2. Discussion of result.
  3. Structure of this work.
  4. Geometric tools
  5. Proof of the lower bound
  6. Proof of the upper bound
  7. Connecting $\mathbf{0}$ and $\mathbf{x}$ to infinite paths in almost surely bounded many steps.
  8. Reaching sufficiently powerful vertices in fewer than $\tfrac{(1+\delta)}{\log\bigl( \frac{\min\{d-k,k\}}{\alpha_k - k}\bigr)}\log\log|x-y|$ many steps, with high probability.
  9. Connecting the two sufficiently powerful vertices through a single connecting vertex with high probability.
  10. Examples

Key Result

Theorem 1.1

In the Poisson-Boolean base model with regularly varying diameters with $u>0$ and $\kappa:= \underset{s\in M}{\operatorname{argmax}} \frac{\min\{d-s,s\}}{\alpha_s -s}$ we have for $x,y\in\mathscr{P}$ and $\delta>0$ in the case $M\neq\emptyset$ and $\alpha_k>k$ for all $k\in\{1,\dots,d\}$ that Furthermore if there exists $k\in\{1,\dots,d\}$ such that $\alpha_k\leq k$ we have that $\operatorname{di

Figures (7)

  • Figure 1: $l_2=\tilde{l}_2+\hat{l}_2$. $\tilde{l}_2=\max\{\tilde{l}_2,\hat{l}_2\}\geq l_2/2$. $B$ with side-lengths at least $l_1/4$ and $l_2/4$.
  • Figure 2: Visualisation of the maximal distance of $B_d$ and $p$ in the directions $e_i$ for $i\in\{1,\dots,d\}$ for the case $d=2$. The blue and red solid lines represent an upper bound for the polytope with diameters $l_1$ and $l_2$. The black lines and hatched green area denote the polytope given by the first $2d$ corners and the diameters of length $l_1,\dots,l_d$. The green rectangle in this convex set is $B_2$ while the rectangle with green boundary and hatched orange is $B_2$ shifted such that it has the maximal distance to $p$, which is given as the pink point.
  • Figure 3: Example for the induction step for the first case of Lemma \ref{['Lemma:one']} from $d=2$ to $d+1=3$ from different perspectives. $B_2$ is in gray, $p$ is the pink point, in green are the connection lines of $p$ and the corners of $B_2$, the orange points as the midpoints of these lines and the orthogonal projection of the midpoints, and $\tilde{B}$ is the box represented by the black mesh.
  • Figure 4: Example for the described induction step of Lemma \ref{['Lemma:one']} from $d=2$ to $d+1=3$ of the second case for $p$. $B_2$ in gray, $p$ the pink point, in turquoise ${z=(2^{-2d+1}l_1,\dots,2^{-2d+1}l_d,2^{-2d+1}l_{d+1})}$, yellow the connection line of $p$ and $y$, in green the connection line of $z$ and $y$, the orange points as $\tilde{N}$ and in purple the boundary of $B$. The hyperpyramid is given as the convex hull of the turquoise point and $B_2$. The boundary of $B_2$ is indicated via the red and blue lines.
  • Figure 5: In cyan blue $\mathbf{0}$ and $\mathbf{x}$, in black the path connecting $\mathbf{0}$ with $\mathbf{z}_0$ (resp. $\mathbf{x}$ with $\mathbf{z}_x$), in blue $\mathbf{z}_0$ and $\mathbf{z}_x$, the path connecting $\mathbf{z}_0$ with $\mathbf{y}_0$ in red (resp. $\mathbf{z}_x$ with $\mathbf{y}_x$), $\mathbf{y}_0$ and $\mathbf{y}_x$ in pink and in orange the vertex $\mathbf{y}$ connecting $\mathbf{y}_0$ and $\mathbf{y}_x$. The dashed gray circles are the boundary of $B_{|x|/8}(0)$, $B_{3|x|/8}(0)$, $B_{|x|/8}(x)$ and $B_{3|x|/8}(x)$. $v_0$ and $v_x$ as defined in \ref{['eq:v0']} and \ref{['eq:vx']}.
  • ...and 2 more figures

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 3.1
  • ...and 6 more