Inhomogeneous bond percolation on square, triangular and hexagonal lattices

TL;DR

This paper uses the star-triangle transformation to analyze inhomogeneous bond percolation on various lattices, establishing criticality, box-crossing inequalities, and extending results to isoradial models, advancing understanding of universality and conformality.

Contribution

It introduces a novel application of the star-triangle transformation to inhomogeneous models, proving criticality and box-crossing inequalities, and extends results to isoradial models.

Findings

Establishes critical points for inhomogeneous bond percolation models.

Derives box-crossing (RSW) inequalities for these models.

Extends proofs to certain isoradial models.

Abstract

The star-triangle transformation is used to obtain an equivalence extending over the set of all (in)homogeneous bond percolation models on the square, triangular and hexagonal lattices. Among the consequences are box-crossing (RSW) inequalities for such models with parameter-values at which the transformation is valid. This is a step toward proving the universality and conformality of these processes. It implies criticality of such values, thereby providing a new proof of the critical point of inhomogeneous systems. The proofs extend to certain isoradial models to which previous methods do not apply.