Bounds of percolation thresholds in the enhanced binary tree

TL;DR

This paper investigates the percolation thresholds of the enhanced binary tree, establishing bounds for the lower threshold and discussing the upper threshold, highlighting the non-exact duality relation due to non-transitivity.

Contribution

It provides new bounds for the lower percolation threshold of the enhanced binary tree and clarifies the relationship between thresholds and duality in non-transitive graphs.

Findings

Lower threshold bounded by 0.355059

Upper threshold bounded by 1/2

Duality relation does not hold exactly

Abstract

By studying its subgraphs, it is argued that the lower critical percolation threshold of the enhanced binary tree (EBT) is bounded as $p_{c1} < 0.355059$, while the upper threshold is bounded both from above and below by 1/2 according to renormalization-group arguments. We also review a correlation analysis in an earlier work, which claimed a significantly higher estimate of $p_{c2}$ than 1/2, to show that this analysis in fact gives a consistent result with this bound. Our result confirms that the duality relation between critical thresholds does not hold exactly for the EBT and its dual, possibly due to the lack of transitivity.