Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The longest edge in discrete and continuous long-range percolation

Arnaud Rousselle, Ercan Sönmez

Abstract

We consider the random connection model in which an edge between two Poisson points at distance $r$ is present with probability $g(r)$. We conduct an extreme value analysis on this model, namely by investigating the longest edge with at least one endpoint within some finite observation window, as the volume of this window tends to infinity. We show that the length of the latter, after normalizing by some appropriate centering and scaling sequences, asymptotically behaves like one of each of the three extreme value distributions, depending on choices of the probability $g(r)$. We prove our results by giving a formal construction of the model by means of a marked Poisson point process and a Poisson coupling argument adapted to this construction. In addition, we study a discrete variant of the model. We obtain parameter regimes with varying behavior in our findings and an unexpected singularity.

The longest edge in discrete and continuous long-range percolation

Abstract

We consider the random connection model in which an edge between two Poisson points at distance is present with probability . We conduct an extreme value analysis on this model, namely by investigating the longest edge with at least one endpoint within some finite observation window, as the volume of this window tends to infinity. We show that the length of the latter, after normalizing by some appropriate centering and scaling sequences, asymptotically behaves like one of each of the three extreme value distributions, depending on choices of the probability . We prove our results by giving a formal construction of the model by means of a marked Poisson point process and a Poisson coupling argument adapted to this construction. In addition, we study a discrete variant of the model. We obtain parameter regimes with varying behavior in our findings and an unexpected singularity.

Paper Structure

This paper contains 14 sections, 16 theorems, 82 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Behavior of the longest edge
  3. Undirected edges
  4. Directed edges
  5. Proofs for the discrete model
  6. Proofs for the continuous model
  7. The typical maximal edge length
  8. A construction of the random connection model
  9. Representation of the exceedances
  10. Estimating $p(x_1,x_2)$
  11. Defining $U_{x_1,x_2}$ and $V_{x_1,x_2}$
  12. Estimating $w(x_1,x_2)$
  13. Concluding
  14. Technical lemmas

Key Result

Theorem 2.1

Let $V=V^{\text{dis}}$. Assume that the function $g$ is given by (F) with the parameter $\alpha \in(d,\infty)$, let ${c}_n= ((\alpha-d)2^{-d}d^{-1}\omega_d^{-1})^\frac{1}{d-\alpha}n^\frac{d}{\alpha-d}$ and $b_n = 0$, $n\in {\mathbb N}$.

Theorems & Definitions (24)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Remark 2.8
  • Proposition 4.1
  • Theorem 4.2: Penrose2
  • ...and 14 more