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Cylinders' percolation: decoupling and applications

Caio Alves, Augusto Teixeira

Abstract

In this paper we establish a strong decoupling inequality for the cylinder's percolation process introduced by Tykesson and Windisch in arXiv:1010.5338 . This model features a very strong dependency structure, making it difficult to study, and this is why such decoupling inequalities are desirable. It is important to notice that the type of dependencies featured by cylinder's percolation is particularly intricate, given that the cylinders have infinite range (unlike some models like Boolean percolation) while at the same time being rigid bodies (unlike processes such as Random Interlacements). Our work introduces a new notion of fast decoupling, proves that it holds for the model in question and finishes with an application. More precisely, we prove that for a small enough density of cylinders, a random walk on a connected component of the vacant set is transient for all dimensions $d \geq 3$.

Cylinders' percolation: decoupling and applications

Abstract

In this paper we establish a strong decoupling inequality for the cylinder's percolation process introduced by Tykesson and Windisch in arXiv:1010.5338 . This model features a very strong dependency structure, making it difficult to study, and this is why such decoupling inequalities are desirable. It is important to notice that the type of dependencies featured by cylinder's percolation is particularly intricate, given that the cylinders have infinite range (unlike some models like Boolean percolation) while at the same time being rigid bodies (unlike processes such as Random Interlacements). Our work introduces a new notion of fast decoupling, proves that it holds for the model in question and finishes with an application. More precisely, we prove that for a small enough density of cylinders, a random walk on a connected component of the vacant set is transient for all dimensions .
Paper Structure (15 sections, 24 theorems, 184 equations, 15 figures)

This paper contains 15 sections, 24 theorems, 184 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Previous results on the model
  3. Overview of the proofs
  4. Open problems
  5. Organization of the paper
  6. A word about constants
  7. Acknowledgments
  8. Preliminaries
  9. The Poisson cylinder process
  10. Decoupling inequalities
  11. Roadmap for the 2-box decoupling inequality
  12. 3-Box decoupling
  13. Renormalization strategy
  14. Efficient unoccupied paths
  15. Finite energy flows

Key Result

Theorem 1.1

There exists a constant $c_{\textnormal{\tiny }c:2boxdec}>0$ depending only on the dimension $d$ such that, for any $\varepsilon, \delta, \alpha \in (0,1)$ and $\rho\in[1,4]$, and any pair of increasing functions if $|x_1 - x_2| \geq L^{2 + \alpha}/\varepsilon$ we have An analogous result for non-increasing functions also holds, see Theorem thm:2boxdec.

Figures (15)

  • Figure 1: Simulations of the occupied ($u = 0.07$) and vacant ($u = 0.07$) sets of the Poisson cylinder process intersected with a ball of radius $24$.
  • Figure 2: Two images showing representations of some of the sets involved in the decoupling inequality.
  • Figure 3: The potentially problematic lines of $\eta$, together with its perturbed versions, $\Gamma_1(\eta)$ and $\Gamma_2(\eta)$. The stochastic operation $\Gamma_i$ consists in fixing the intersection point of a line with the plane $\Pi_i$ and resampling its direction, conditioned on it being "problematic".
  • Figure 4: By enlarging the radii of the perturbed cylinders, we will have the required domination between the cylinder processes: indeed the intersection of the smaller cylinders with the box $B_!$ will be contained in the intersection of the larger cylinders with the same box.
  • Figure 5: The figure shows schematically how $\Gamma_1$ acts on a line $l$.
  • ...and 10 more figures

Theorems & Definitions (53)

  • Theorem 1.1
  • Remark 1
  • Theorem 1.2
  • Remark 2
  • Lemma 2.1
  • Theorem 3.1
  • Remark 3
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 43 more