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Invariant splitting of a slab into infinitely many robust clusters

Péter Mester

Abstract

We give an example of an invariant bond percolation process on the slab $\mathbb{Z}^2\times \{0,1\}$ with the property that it has infinitely many clusters whose critical percolation probability is strictly less than $1$. We also show that no such process can exist in $\mathbb{Z}^2$.

Invariant splitting of a slab into infinitely many robust clusters

Abstract

We give an example of an invariant bond percolation process on the slab with the property that it has infinitely many clusters whose critical percolation probability is strictly less than . We also show that no such process can exist in .
Paper Structure (8 sections, 3 theorems, 6 equations, 8 figures)

This paper contains 8 sections, 3 theorems, 6 equations, 8 figures.

Table of Contents

  1. Introduction
  2. A visual guide to ${\tt RF}$
  3. A modification of ${\tt RF}$ to define it through a graph union
  4. One-ended trees
  5. Cutting and folding
  6. Crossing probabilities
  7. The case of ${\mathbb Z}^2$
  8. Acknowledgements

Key Result

Theorem 1.1

There is an invariant bond process $\Phi_{\beta}({\tt RF})$ on the planar slab ${\mathbb Z}^2\times \{0,1\}$ which has infinitely many robust clusters.

Figures (8)

  • Figure 1.1: The shaded part of the left picture is ${\tt RF}$, within which the dark region illustrates a "road". The right picture illustrates that if a path exists between the opposite sides of each rectangle (in an appropriately alternating way between vertical and horizontal ones), then we can extract an infinite path from them. If the rectangles are getting "thicker and thicker", then in a Bernoulli $p$ percolation ($p\in (1/2,1)$) this happens with positive probability due to the lower bound on the crossing probability (Lemma \ref{['l.BRcorr']} below).
  • Figure 1.2: Each of the three vertical slices named $AV, BV , CV$ is meant to intersect only the horizontal slice ($AH,BH, CH$) with which its first letter agrees. In the extra space we have in the slab, we can fold each vertical slice in a way that the folded slices keep the intersections we wanted but all other intersections are avoided.
  • Figure 2.1: A piece of a $(2,3)$-grid.
  • Figure 2.2: The left picture shows a nested sequence of windows. The right one illustrates the fork (top left) and extended fork (bottom-left) associated to a given window in a given sequence of nested windows. The vertical and horizontal frames included in the extended forks of all the windows will constitute the sets ${\tt Rect}_{\tt V}$ and ${\tt Rect}_{\tt H}$.
  • Figure 5.1: Within the slab we can fold the slices (as if they were ribbons) so that only the ones of the same shade intersect.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 6.1