# On a $p$-Adic Generalized Gibbs Measure for Ising Model on a Cayley Tree

## TL;DR

This paper investigates $p$-adic Ising models on Cayley trees, providing a complete description of translation-invariant Gibbs measures for a specific case and demonstrating phase transitions for certain primes and tree orders.

## Contribution

It offers a full characterization of $p$-adic Gibbs measures for the Ising model on Cayley trees and establishes conditions for phase transitions based on prime congruences.

## Key findings

- Complete description of translation-invariant Gibbs measures for $k=3$
- Existence of phase transition for $p$-adic Ising models when $p \\equiv 1 \\operatorname{mod }4$ and $k \\geq 3$
- Phase transition phenomena depend on prime congruences and tree order

## Abstract

In this paper we consider a $p$-adic Ising model on the Cayley tree of order $k\geq 2$. We give full description of all $p$-adic translation-invariant generalized Gibbs measures for $k=3$. Moreover, we show the existence of phase transition for $p$-adic Ising model for any $k\geq3$ when $p\equiv1(\operatorname{mod }4)$.