Critical percolation on mesoscopic triangulations

TL;DR

This paper extends the proof of conformal invariance for critical percolation from the triangular lattice to more complex periodic graphs, demonstrating a form of universality in percolation behavior.

Contribution

It generalizes Smirnov's proof to a broader class of graphs with large-scale structures, beyond the regular triangular lattice.

Findings

Conformal invariance holds for certain periodic graphs.

The result suggests a weak form of universality in critical percolation.

The approach broadens understanding of percolation on complex structures.

Abstract

We extend Smirnov's proof of the existence and conformal invariance of the scaling limit of critical site-percolation on the triangular lattice to particular sequences of periodic graphs with more arbitrary large-scale structure, obtained by piecing together triangular regions of the triangular lattice. While not formally speaking a scaling limit statement (as the graphs are not rescaled versions of each other), the result is a weak form of universality for critical percolation.