Reply to Comment on `Monte-Carlo simulation study of the two-stage percolation transition in enhanced binary trees'

TL;DR

This paper clarifies the nature of the two-stage percolation transition in enhanced binary trees, addressing previous disagreements by analyzing the limitations of finite size scaling on nonamenable structures.

Contribution

It provides a detailed discussion on the proper analysis of percolation transitions in nonamenable graphs, correcting prior misestimations.

Findings

Finite size scaling is inadequate for nonamenable trees.

Numerical results support the revised understanding of the transition.

Clarifies the disagreement in previous transition probability estimates.

Abstract

We discuss the nature of the two-stage percolation transition on the enhanced binary tree in order to explain the disagreement in the estimation of the second transition probability between the one in our recent paper (J. Phys. A:Math. Theor. 42 (2009) 145001) and the one in the comment to it from Baek, Minnhagen and Kim. We point out some reasons that the finite size scaling analysis used by them is not proper for the enhanced tree due to its nonamenable nature, which is verified by some numerical results.