Constrained percolation, Ising model, and XOR Ising model on planar lattices. Issue 2 (7th May 2020)

• Record Type: Journal Article
• Authors: Li, Zhongyang
• Is Part Of: Random structures & algorithms. Volume 57:Issue 2(2020)
• Page Start: 474
• Page End: 525
• Publication Date: 2020-05-07
• Subjects: Percolation -- XOR Ising -- Planar graph; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20924
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract : We study site percolation models on planar lattices including the [ m, 4, n, 4] lattice and the square tilings on the Euclidean plane ( R 2 ) or the hyperbolic plane ( H 2 ), satisfying certain local constraints on degree‐4 faces. These models are closely related to Ising models and XOR Ising models (product of two i.i.d Ising models) on regular tilings of R 2 or H 2 . In particular, we obtain a description of the numbers of infinite "+" and "−" clusters of the ferromagnetic Ising model on a vertex‐transitive triangular tiling of H 2 for different boundary conditions and coupling constants. Our results show the possibility that such an Ising configuration has infinitely many infinite "+" and "−" clusters, while its random cluster representation has no infinite open clusters. Percolation properties of corresponding XOR Ising models are also discussed.