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Random interlacements on transient weighted graphs: 0-1 laws and FKG inequality

Orphée Collin

Abstract

We study some properties of the random interlacement model on a transient weighted graph, which was introduced by A. Teixeira in ["Interlacement percolation on transient weighted graphs", Augusto Teixeira, Electronic Journal of Probability (2009)]. We give a simple proof of the FKG-property and discuss the occurrence of several 0-1 laws for non-local events. We show in particular a 0-1 law for some increasing non-local events, without any assumption.

Random interlacements on transient weighted graphs: 0-1 laws and FKG inequality

Abstract

We study some properties of the random interlacement model on a transient weighted graph, which was introduced by A. Teixeira in ["Interlacement percolation on transient weighted graphs", Augusto Teixeira, Electronic Journal of Probability (2009)]. We give a simple proof of the FKG-property and discuss the occurrence of several 0-1 laws for non-local events. We show in particular a 0-1 law for some increasing non-local events, without any assumption.
Paper Structure (25 sections, 17 theorems, 109 equations)

This paper contains 25 sections, 17 theorems, 109 equations.

Table of Contents

  1. Overview
  2. The model
  3. Some notations
  4. Interlacement process
  5. Theorems
  6. FKG inequality
  7. The sigma-algebra of non-local events
  8. Under tail-triviality or tail-atomicity assumption
  9. Quantitative criterion
  10. A weak 0-1 law
  11. A 0-1 law for increasing non-local events
  12. Preliminaries for the proof of the 0-1 laws
  13. Hinge decomposition of the interlacement process
  14. Coupling results
  15. Proofs under assumptions on TMC
  16. ...and 10 more sections

Key Result

Lemma 2

Let $(\mathcal{X},\mathcal{F} , \le)$ and $(\mathcal{Y}, \mathcal{G} , \le)$ be two mesurable spaces, equiped with orders. Let $X$ be an $\mathcal{X}$-valued random variable and let $f:\mathcal{X} \to \mathcal{Y}$ be a measurable and non-decreasing function. If $X$ satifies the FKG inequality, then

Theorems & Definitions (42)

  • Remark 1
  • Lemma 2
  • Theorem 3
  • proof
  • Example 4
  • Theorem 5
  • Theorem 6
  • Remark 7
  • Theorem 8
  • Remark 9
  • ...and 32 more