On Gibbs Measures of $P$-Adic Potts Model on the Cayley Tree

Farrukh Mukhamedov, Utkir Rozikov

Published 2005-10-06, updated 2006-02-25Version 2

We consider a nearest-neighbor $p$-adic Potts (with $q\geq 2$ spin values and coupling constant $J\in \Q_p$) model on the Cayley tree of order $k\geq 1$. It is proved that a phase transition occurs at $k=2$, $q\in p\mathbb{N}$ and $p\geq 3$ (resp. $q\in 2^2\mathbb{N}$, $p=2$). It is established that for $p$-adic Potts model at $k\geq 3$ a phase transition may occur only at $q\in p\mathbb{N}$ if $p\geq 3$ and $q\in 2^2\mathbb{N}$ if $p=2$.