Topological Percolation on Hyperbolic Simplicial Complexes

TL;DR

This paper develops a topological percolation theory for hyperbolic simplicial complexes in dimensions 2 and 3, revealing multiple phase transitions and complex critical behaviors, including a BKT transition in 3D.

Contribution

It introduces a comprehensive topological percolation framework for hyperbolic simplicial complexes, extending understanding beyond traditional node and link percolation.

Findings

Identification of four percolation problems in 2D complexes

Discovery of six percolation problems in 3D complexes

Observation of a BKT transition in 3D triangle percolation

Abstract

Simplicial complexes are increasingly used to understand the topology of complex systems as different as brain networks and social interactions. It is therefore of special interest to extend the study of percolation to simplicial complexes. Here we propose a topological theory of percolation for discrete hyperbolic simplicial complexes. Specifically we consider hyperbolic manifolds in dimension $d=2$ and $d=3$ formed by simplicial complexes, and we investigate their percolation properties in the presence of topological damage, i.e., when nodes, links, triangles or tetrahedra are randomly removed. We show that in $d=2$ simplicial complexes there are four topological percolation problems and in $d=3$, there are six. We demonstrate the presence of two percolation phase transitions characteristic of hyperbolic spaces for the different variants of topological percolation. While most of the known results on percolation in hyperbolic manifolds are in $d=2$, here we uncover the rich critical behavior of $d=3$ hyperbolic manifolds, and show that triangle percolation displays a Berezinskii-Kosterlitz-Thouless (BKT) transition. Finally we provide evidence that topological percolation can display a critical behavior that is unexpected if only node and link percolation are considered.