Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Disks, Surfaces, and Entanglement Percolation

Paul Duncan, Benjamin Schweinhart, David Sivakoff

TL;DR

This work analyzes the topology of plaquette percolation on $\mathbb{Z}^3$, focusing on when large loops bound disks (contractibility) and how this relates to entanglement percolation. By introducing a tractable, limited notion of entanglement ($m$-entanglement) and a duality with the primal plaquette model, the authors establish a robust area-law regime below a truncation threshold and a perimeter-law regime above an entanglement threshold, with a nontrivial intermediate regime where surfaces acquire many handles. They prove continuity of $m$-entanglement thresholds in slabs and develop a surgical topological framework to show that large plaquette crossings typically generate substantial first homology, revealing rich intermediate-topology phenomena. The results extend the classical percolation paradigm to higher dimensions, connecting homotopy, homology, and nonlocal entanglement in a precise, quantifiable way.

Abstract

We study the probability that a loop is null-homotopic -- that is, bounded by the continuous image of a disk -- in plaquette percolation on $\mathbb{Z}^3.$ Locally, the event that there is a ``horizontal disk crossing'' of a rectangular prism is dual to the event that there is a vertical crossing in entanglement percolation (with wired boundary conditions). However, the analysis of analogous events on the full lattice is complicated by the long-range nature of entanglement percolation. We show that the probability that a rectangular loop is contractible exhibits a phase transition from area law to perimeter law dual to the entanglement percolation threshold, conditional on a conjecture concerning the continuity of entanglement percolation thresholds with respect to truncation. We also show the continuity of a truncated entanglement percolation threshold in slabs and apply that to identify a regime where large plaquette surfaces exist but typically have many handles.

Disks, Surfaces, and Entanglement Percolation

TL;DR

This work analyzes the topology of plaquette percolation on , focusing on when large loops bound disks (contractibility) and how this relates to entanglement percolation. By introducing a tractable, limited notion of entanglement (-entanglement) and a duality with the primal plaquette model, the authors establish a robust area-law regime below a truncation threshold and a perimeter-law regime above an entanglement threshold, with a nontrivial intermediate regime where surfaces acquire many handles. They prove continuity of -entanglement thresholds in slabs and develop a surgical topological framework to show that large plaquette crossings typically generate substantial first homology, revealing rich intermediate-topology phenomena. The results extend the classical percolation paradigm to higher dimensions, connecting homotopy, homology, and nonlocal entanglement in a precise, quantifiable way.

Abstract

We study the probability that a loop is null-homotopic -- that is, bounded by the continuous image of a disk -- in plaquette percolation on Locally, the event that there is a ``horizontal disk crossing'' of a rectangular prism is dual to the event that there is a vertical crossing in entanglement percolation (with wired boundary conditions). However, the analysis of analogous events on the full lattice is complicated by the long-range nature of entanglement percolation. We show that the probability that a rectangular loop is contractible exhibits a phase transition from area law to perimeter law dual to the entanglement percolation threshold, conditional on a conjecture concerning the continuity of entanglement percolation thresholds with respect to truncation. We also show the continuity of a truncated entanglement percolation threshold in slabs and apply that to identify a regime where large plaquette surfaces exist but typically have many handles.

Paper Structure

This paper contains 9 sections, 32 theorems, 64 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Outline
  4. Limited Entanglement and Entanglement Duality
  5. Topological Subtleties
  6. Separating Spheres
  7. Area Law
  8. Continuity of $m$-entanglement in slabs
  9. Many Handles in the Intermediate Regime

Key Result

Proposition 1

Let $\gamma$ be a loop in $\mathbb{Z}^3.$ Then $U_{\gamma}$ occurs if and only if there is no set of dual bonds $E \subset B$ so that $E \cup \gamma$ is entangled.

Figures (8)

  • Figure 1: Plaquette events: (left) the loop $\gamma$ (shown in black) is null-homotopic as it is bounded by an orange surface of plaquettes homeomorphic to a disk. (right) The loop $\gamma$ is null-homologous (bounded by a surface) but not null-homotopic. In fact, a null-homotopy is precluded by the existence of two loops of dual bonds (shown in blue) which, together with $\gamma$, form a set of Borromean rings.
  • Figure 2: Two views of a plaquette crossing of $\Lambda_{25}$ in plaquette percolation with probability $p=1-0.246\approx 1-p_c\left( \mathbb{Z}^3 \right)+.003,$ chosen among the crossings with a minimal number of plaquettes. Note the presence of handles. The figure was created by Benjamin Atelsek, Gregory Maleski, and Tristan Napoliello as part of a project run through the Mason Experimental Geometry Lab (MEGL).
  • Figure 3: An illustration of the event in Proposition \ref{['prop:duality']}, shown from two different perspectives. $D_2$ is shown in blue on the left and $D_1$ in green on the right. Observe that $\pi_1\left( D_2 \right)=\mathbb{Z}$ and that the generator of that fundamental group is not contractible in the union of $D_2$ and the orange surface due to the presence of a handle. The black paths are an entangled crossing of $D_1.$
  • Figure 4: An example inspired by Bing's "House with Two Rooms"bing1964some, shown from two different perspectives. Here, $P_R$ is depicted in orange, $D_1$ is illustrated in blue, and a system of bonds homotopy equivalent to $B_R$ is drawn in black. $P_R$ is formed from three parallel rectangles by removing a disk from the top and bottom rectangles and two disks from the middle rectangle and connecting the middle rectangle to the other two by cylinders. It is easy to see that $D_1$ is separated by a sphere in the complement of $B_R,$ but it is more difficult to visualize a disk inside of $R\setminus B_R$ separating $D_1.$ If $\alpha_1, \alpha_2,$ and $\alpha_3$ are the three rectangular loops in $D_2\cap P_R,$ ordered from top to bottom, then $\left[ \alpha_2 \right]=\left[ \alpha_1 \right]\ast\left[ \alpha_3 \right]$ in $\pi_1\left( P_R \right),$ but $\left[ \alpha_1 \right]=\left[ \alpha_2 \right]=\left[ \alpha_3 \right]$ in $\pi_1\left( D_2 \right).$ It follows that $\left[ \alpha_1 \right]=e$ in $\pi_1\left( P_R\cup D_2 \right).$ Then, Dehn's lemma implies that $\alpha_1$ is the boundary of a disk in a small neighborhood of $D_2\cup P_R,$ which can then be pushed inside of $R\setminus B_R.$ This example will also be relevant in Section \ref{['sec:genus']}.
  • Figure 5: An illustration of each of the Reidemeister moves with the addition of a strand from $\gamma$ in red in several possible configurations.
  • ...and 3 more figures

Theorems & Definitions (52)

  • Proposition 1
  • Theorem 2
  • Proposition 3
  • Theorem 4
  • Lemma 5
  • proof
  • Theorem 6: Dehn's Lemma papakyriakopoulos1957dehn
  • Theorem 7: Bing bing1957approximating
  • Theorem 8: Alexander alexander1924subdivision
  • Corollary 9
  • ...and 42 more