Constrained percolation in two dimensions

TL;DR

This paper proves the absence of infinite clusters and contours in certain critical constrained percolation models on the square lattice, using new combinatorial techniques and planar duality, with applications to dimer and XOR Ising models.

Contribution

It introduces novel combinatorial methods to analyze constrained percolation, establishing key properties like absence of infinite clusters and contours under weak conditions.

Findings

No infinite clusters in critical constrained percolation models.

At most one infinite contour in high-temperature XOR Ising models.

No infinite contour in low-temperature XOR Ising models.

Abstract

We prove absence of infinite clusters and contours in a class of critical constrained percolation models on the square lattice. The percolation configuration is assumed to satisfy certain hard local constraints, but only weak symmetry and ergodicity conditions are imposed on its law. The proofs use new combinatorial techniques exploiting planar duality. Applications include absence of infinite clusters of diagonal edges for critical dimer models on the square-octagon lattice, as well as absence of infinite contours and infinite clusters for critical XOR Ising models on the square grid. We also prove that there exists at most one infinite contour for high-temperature XOR Ising models, and no infinite contour for low-temperature XOR Ising model.