Disks, Surfaces, and Entanglement Percolation

TL;DR

This paper investigates phase transitions in entanglement percolation on 3D lattices, linking topological properties of loops to percolation thresholds and exploring the existence of complex surfaces with multiple handles.

Contribution

It establishes a connection between loop contractibility probabilities and entanglement percolation thresholds, introducing new insights into phase transitions and surface topology in 3D percolation models.

Findings

Phase transition from area law to perimeter law in loop contractibility

Continuity of entanglement percolation thresholds in slabs

Existence of large, complex plaquette surfaces with many handles

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.