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Decay of connection probability in high-dimensional continuum percolation

Matthew Dickson, Yucheng Liu

Abstract

We study a percolation model on $\mathbb R^d$ called the random connection model. For $d$ large, we use the lace expansion to prove that the critical two-point connection probability decays like $|x|^{-(d-2)}$ as $|x| \to \infty$, with possible anisotropic decay. Our proof also applies to nearest-neighbour Bernoulli percolation on $\mathbb Z^d$ in $d \ge 11$ and simplifies considerably the proof given by Hara in 2008. The method is based on the recent deconvolution strategy of Liu and Slade and uses an $L^p$ version of Hara's induction argument.

Decay of connection probability in high-dimensional continuum percolation

Abstract

We study a percolation model on called the random connection model. For large, we use the lace expansion to prove that the critical two-point connection probability decays like as , with possible anisotropic decay. Our proof also applies to nearest-neighbour Bernoulli percolation on in and simplifies considerably the proof given by Hara in 2008. The method is based on the recent deconvolution strategy of Liu and Slade and uses an version of Hara's induction argument.

Paper Structure

This paper contains 22 sections, 15 theorems, 131 equations, 2 figures.

Table of Contents

  1. Introduction and results
  2. The model and main result
  3. Strategy of proof
  4. Organisation
  5. Diagrammatic estimates
  6. Diagrams
  7. Proof of Lemma \ref{['lem:Pi_base']}
  8. Proof of Proposition \ref{['prop:climb']}
  9. Bound on diagrams
  10. Fourier transform estimates
  11. Proof of Lemma \ref{['lem:simple']}
  12. Edge
  13. Triangle
  14. Bubble
  15. Martini
  16. ...and 7 more sections

Key Result

Theorem 1.3

Let $\varphi$ obey Assumption ass:adj and let $a_d = \frac{ \Gamma(\frac{ d-2 }{2} ) }{ 2\pi^{d/2}}$. If $d > d_0$ with $d_0 \ge 8$ sufficiently large, then there is a positive-definite diagonal matrix $\Sigma$ such that The matrix $\Sigma$ is given explicitly in terms of lace expansion quantities in eq:Sigma.

Figures (2)

  • Figure 1: Graphical notation for the $\psi$ functions. Each $\bullet$ has a vertex factor $\lambda$.
  • Figure 2: Diagrams of the $\bar{\psi}_0$, and $\bar{\psi}$ functions.

Theorems & Definitions (42)

  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 1.5
  • Proposition 1.6
  • Lemma 1.7
  • proof : Proof of Theorem \ref{['thm:main']}
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • ...and 32 more