Sharpness of the phase transition for the corrupted compass model on transitive graphs

Beekenkamp, Thomas (2021): Sharpness of the phase transition for the corrupted compass model on transitive graphs. In: Indagationes Mathematicae-New Series, Vol. 32, No. 3: pp. 736-744

Abstract

In the corrupted compass model on a vertex-transitive graph, a neighbouring edge of every vertex is chosen uniformly at random and opened. Additionally, with probability p, independently for every vertex, each of its neighbouring edges is opened. We study the size of open clusters in this model. Hirsch et al. (2018) have shown that for small p all open clusters are finite almost surely, while for large p, depending on the underlying graph, there exists an infinite open cluster almost surely. We show that the corresponding phase transition is sharp, i.e., in the subcritical regime, all open clusters are exponentially small. Furthermore we prove a mean-field lower bound in the supercritical regime. The proof uses the by now well established method based on the OSSS inequality. A second goal of this note is to showcase this method in an uncomplicated setting. ?c 2021 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

• Item Type: Journal article
• Faculties: Mathematics, Computer Science and Statistics > Mathematics
• Subjects: 500 Science > 510 Mathematics
• ISSN: 0019-3577
• Language: English