Abstract
A uniqueness condition for Gibbs measures is given. This condition is stated in terms of (absence of) a certain type of percolation involving two independent realisations. This result can be applied in certain concrete situations by comparison with “ordinary” percolation. In this way we prove that the Ising antiferromagnet on a square lattice has a unique Gibbs measure if β(4−|h⥻)<1/2ln(P c /(1−P c )), whereh denotes the external magnetic field, β the inverse temperature, andP c the critical probability for site percolation on that lattice. SinceP c is larger than 1/2, this extends a result by Dobrushin, Kolafa and Shlosman (whose proof was computer-assisted).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.