On Uniqueness of Gibbs Measures for $P$-Adic Nonhomogeneous $\l$-Model on the Cayley Tree

TL;DR

This paper investigates the conditions under which a $p$-adic $$-model on a Cayley tree has a unique Gibbs measure, showing no phase transition for $p \u2265 3$ and uniqueness for the $p$-adic Ising model.

Contribution

It establishes the existence and uniqueness conditions for $p$-adic Gibbs measures on Cayley trees, including specific results for the $p$-adic Ising model.

Findings

No phase transition for $p  3$

Unique $p$-adic Gibbs measure for the $p$-adic Ising model

Bounded Gibbs measure if and only if $p  3$

Abstract

We consider a nearest-neighbor $p$-adic $\l$-model with spin values $\pm 1$ on a Cayley tree of order $k\geq 1$. We prove for the model there is no phase transition and as well as the unique $p$-adic Gibbs measure is bounded if and only if $p\geq 3$. If $p=2$ then we find a condition which guarantees nonexistence of a phase transition. Besides, the results are applied to the $p$-adic Ising model and we show that for the model there is a unique $p$-adic Gibbs measure.