A homological upper bound on critical probabilities for hyperbolic percolation

TL;DR

This paper establishes an upper bound on the critical probabilities for hyperbolic percolation by linking percolation phenomena to homology in finite graph approximations, advancing understanding of phase transitions in hyperbolic structures.

Contribution

It introduces a novel homological approach to bounding critical probabilities in hyperbolic percolation models, connecting topological properties with probabilistic thresholds.

Findings

Derived an explicit upper bound on critical probabilities for hyperbolic graphs.

Linked percolation thresholds to the emergence of homology in finite graph models.

Provided new insights into phase transition behavior in hyperbolic percolation.

Abstract

We study bond percolation for a family of infinite hyperbolic graphs. We relate percolation to the appearance of homology in finite versions of these graphs. As a consequence, we derive an upper bound on the critical probabilities of the infinite graphs.