Constrained percolation, Ising model and XOR Ising model on planar lattices

TL;DR

This paper investigates constrained percolation, Ising, and XOR Ising models on planar lattices, revealing complex cluster behaviors and phase transitions, especially on hyperbolic and Euclidean tilings, with implications for understanding infinite cluster existence.

Contribution

It provides a comprehensive analysis of infinite cluster configurations in constrained percolation and related Ising models on various planar lattices, including hyperbolic tilings, highlighting new phenomena and behaviors.

Findings

Complete characterization of infinite cluster counts in hyperbolic Ising models.

Existence of multiple infinite clusters under certain conditions.

No infinite clusters in critical random cluster models on non-amenable graphs.

Abstract

We study constrained percolation models on planar lattices including the $[m,4,n,4]$ lattice and the square tilings of the hyperbolic plane, satisfying certain local constraints on faces of degree 4, and investigate the existence of infinite clusters. The constrained percolation models on these lattices are closely related to Ising models and XOR Ising models on regular tilings of the Euclidean plane or the hyperbolic plane. In particular, we obtain a complete picture of the number of infinite "$+$" and "$-$" clusters of the ferromagnetic Ising model with the free boundary condition on a vertex-transitive triangular tiling of the hyperbolic plane with all the possible values of coupling constants. Our results show that for the Ising model on a vertex-transitive triangular tiling of the hyperbolic plane, it is possible that its random cluster representation has no infinite open clusters, while the Ising model has infinitely many infinite "$+$"-clusters and infinitely many infinite "$-$"-clusters. We also study different behaviors the infinite "$+$" and "$-$" clusters of XOR Ising models on regular tilings of the Euclidean plane and the hyperbolic plane for different coupling constants. A by-product we prove is the result that the critical random cluster model with $q\geq 1$ and the wired boundary condition on a quasi-transitive, non-amenable, unimodular graph almost surely has no infinite open clusters.