Continuum Percolation on Disoriented Surfaces: the Problem of Permeable Disks on a Klein Bottle

TL;DR

This study investigates continuum percolation on Klein bottles, revealing topological invariance of the percolation threshold and scaling exponents, while showing wrapping probabilities depend on surface topology.

Contribution

It demonstrates that the percolation threshold and scaling exponents are topologically invariant, and highlights the topological dependence of wrapping probabilities on Klein bottles versus tori.

Findings

Percolation threshold on Klein bottle matches that on torus.

Scaling exponents are topologically invariant.

Wrapping probabilities differ between Klein bottle and torus.

Abstract

The percolation threshold and wrapping probability $R_{\infty}$ for the two-dimensional problem of continuum percolation on the surface of a Klein bottle have been calculated by the Monte Carlo method with the Newman--Ziff algorithm for completely permeable disks. It has been shown that the percolation threshold of disks on the Klein bottle coincides with the percolation threshold of disks on the surface of a torus, indicating that this threshold is topologically invariant. The scaling exponents determining corrections to the wrapping probability and critical concentration owing to the finite-size effects are also topologically invariant. At the same time, the quantities $R_{\infty}$ are different for percolation on the torus and Klein bottle and are apparently determined by the topology of the surface. Furthermore, the difference between the $R_{\infty}$ values for the torus and Klein bottle means that at least one of the percolation clusters is degenerate.