Translation-invariant generalized $P$-adic Gibbs measures for the
Ising model on Cayley trees
Abstract
Main aim of the present paper is explore certain physical phenomena by
means of $p$-adic probability theory. To overcome this study, we deal
with a more general setting to define $p$-adic Gibbs measures. For the
sake of simplicity of explanations, we restrict ourselves to the Ising
model on the Cayley tree, since such a model has broad theoretical and
practical applications. To study $p$-adic quasi Gibbs measures, we
reduce the problem to the description of the fixed points of the
Ising-Potts mapping. Finding fixed points is not an easy job as in the
real setting. Furthermore, the phase transition for the model is
established. In the real case, the phase transition yields the the
singularity of the limiting Gibbs measures. However, we show that the
$p$-adic quasi Gibbs measures do not exhibit the mentioned type of
singularity, such kind of phenomena is called strong phase transition.
Finally, we deal with the solvability and the number of solutions of
ceratin $p$-adic equation depending on several parameters. Such a
description allows us to find all possible translation-invariant
$p$a-adic quasi Gibbs measures.