Hierarchical renormalization-group study on the planar bond-percolation problem

TL;DR

This paper applies a hierarchical renormalization-group approach to analyze planar bond-percolation, providing bounds and estimates for critical probabilities and the correlation-length exponent in two-dimensional lattices.

Contribution

It extends hierarchical renormalization-group techniques to 2D planar lattices, offering bounds and estimates for percolation critical parameters.

Findings

Lower bounds and exact critical probabilities obtained

Correlation-length critical exponent estimated as approximately 1

Method applicable to hierarchical and planar structures

Abstract

For certain hierarchical structures, one can study the percolation problem using the renormalization-group method in a very precise way. We show that the idea can be also applied to two-dimensional planar lattices by regarding them as hierarchical structures. Either a lower bound or an exact critical probability can be obtained with this method and the correlation-length critical exponent is approximately estimated as $\nu \approx 1$.