The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices

TL;DR

This paper derives a conditional formula for the critical percolation manifold of inhomogeneous bow-tie lattices, extending the class of solvable systems and providing evidence for broader applicability through rigorous and conjectured arguments.

Contribution

It introduces a new conditional derivation of the inhomogeneous critical manifold for bow-tie lattices, expanding solvable models and connecting to checkerboard lattice results.

Findings

Derived the critical percolation manifold formula for bow-tie lattices.

Extended solvable class to include certain inhomogeneous and asymmetric lattices.

Provided evidence linking the formula to checkerboard lattice results.

Abstract

We give a conditional derivation of the inhomogeneous critical percolation manifold of the bow-tie lattice with five different probabilities, a problem that does not appear at first to fall into any known solvable class. Although our argument is mathematically rigorous only on a region of the manifold, we conjecture that the formula is correct over its entire domain, and we provide a non-rigorous argument for this that employs the negative probability regime of the triangular lattice critical surface. We discuss how the rigorous portion of our result substantially broadens the range of lattices in the solvable class to include certain inhomogeneous and asymmetric bow-tie lattices, and that, if it could be put on a firm foundation, the negative probability portion of our method would extend this class to many further systems, including F.Y. Wu's checkerboard formula for the square lattice. We conclude by showing that this latter problem can in fact be proved using a recent result of Grimmett and Manolescu for isoradial graphs, lending strong evidence in favour of our other conjectured results.