On weakly periodic Gibbs measures for the Potts model with external field on the Cayley tree

Abstract

We study the Potts model with external field on the Cayley tree of order $k \geq 2$. For the antiferromagnetic Potts model with external field and $k \geq 6$ and $q \geq 3$, it is shown that the weakly periodic Gibbs measure, which is not periodic, is not unique. For the Potts model with external field equal to zero, we also study weakly periodic Gibbs measures. It is shown that, under certain conditions, the number of these measures cannot be smaller than $2^q - 2$.