New weakly periodic Gibbs measures of Ising model on Cayley tree
The article considers the Ising model on a regular tree \(T=(V,E)\) of degree \(k+1\), under the presence of a magnetic field \(h=(h_x)_{x\in V}\). Given the field, one can define Ising measures in finite volume, \(\mu_n\), supported on the ball \(V_n\) of radius \(n\) (without boundary condition), and they satisfy concordance if the marginal of \(\mu_n\) on \(V_{n-1}\) coincides with \(\mu_{n-1}\); in this case, one can extend the \(\mu_n\) into an infinite-volume Gibbs measure. Classifying the Gibbs measures obtained using the above construction and that are invariant under the action of a subgroup \(H\) of automorphism is then a natural question. The concordance can be rewritten into compatibility relations satisfied by the field \(h\) at each vertex, and which depend on the temperature. The invariance of \(\mu\) under the action of \(H\) then transforms into that of \(h\), and the question becomes the existence of solutions to the compatibility relations that are themselves invariant under \(H\). The article studies that question, and constructs such invariant solutions (and the associated Gibbs measures) in the case where \(H\) is of index \(2\).
scientific article; zbMATH DE number 6534154