[Paper Review] On $p$-adic Gibbs Measures for Hard Core Model on a Cayley Tree

Daniel Gandolfo, U. A. Rozikov+1 more|arXiv (Cornell University)|Jul 25, 2011

advanced mathematical theories21 references22 citations

TL;DR

This paper investigates $p$-adic Gibbs measures for the hard core model on a Cayley tree, introducing a non-Archimedean framework that reveals stark contrasts with classical real-valued models. It proves that $p$-adic splitting Gibbs measures exist only if $p$ divides $2^k - 1$, and for $k=2$, such measures exist precisely when $p=3$, yielding two distinct periodic non-translation-invariant measures for $\lambda$ in a $p$-adic ball of radius $1/27$, with boundedness occurring iff $p \neq 3$. The results highlight fundamental structural differences between $p$-adic and real Gibbs measures due to the non-Archimedean norm and restricted domain of $p$-adic exponential functions.

ABSTRACT

In this paper we consider a nearest-neighbor $p$-adic hard core (HC) model, with fugacity $λ$, on a homogeneous Cayley tree of order $k$ (with $k + 1$ neighbors). We focus on $p$-adic Gibbs measures for the HC model, in particular on $p$-adic "splitting" Gibbs measures generating a $p$-adic Markov chain along each path on the tree. We show that the $p$-adic HC model is completely different from real HC model: For a fixed $k$ we prove that the $p$-adic HC model may have a splitting Gibbs measure only if $p$ divides $2^k-1$. Moreover if $p$ divides $2^k-1$ but does not divide $k+2$ then there exists unique translational invariant $p$-adic Gibbs measure. We also study $p$-adic periodic splitting Gibbs measures and show that the above model admits only translational invariant and periodic with period two (chess-board) Gibbs measures. For $p\geq 7$ (resp. $p=2,3,5$) we give necessary and sufficient (resp. necessary) conditions for the existence of a periodic $p$-adic measure. For k=2 a $p$-adic splitting Gibbs measures exists if and only if p=3, in this case we show that if $λ$ belongs to a $p$-adic ball of radius 1/27 then there are precisely two periodic (non translational invariant) $p$-adic Gibbs measures. We prove that a $p$-adic Gibbs measure is bounded if and only if $p e 3$.

Motivation & Objective

• To investigate the existence and structure of $p$-adic Gibbs measures for the hard core model on a Cayley tree, a model previously studied only over the reals.
• To identify the conditions under which $p$-adic splitting Gibbs measures exist, particularly focusing on translational invariance and periodicity.
• To compare the behavior of $p$-adic Gibbs measures with their classical real counterparts, highlighting fundamental differences due to the non-Archimedean nature of $p$-adic numbers.
• To analyze the boundedness of $p$-adic Gibbs measures and determine the precise conditions under which they are bounded or unbounded.

Proposed method

• The study employs $p$-adic analysis, using the non-Archimedean $p$-adic norm and the strong triangle inequality to analyze measure behavior.
• It formulates a functional equation for $p$-adic Gibbs measures based on the hard core interaction and Cayley tree structure.
• The existence of splitting Gibbs measures is analyzed via number-theoretic conditions on $p$ and $k$, particularly $p \mid 2^k - 1$.
• Periodic Gibbs measures are studied by imposing periodicity conditions (e.g., period two, or chess-board), reducing the problem to solving systems of equations over $\mathbb{Q}_p$.
• The boundedness of the measure is analyzed using the $p$-adic norm of the measure's values, particularly through the behavior of $|\mu_n|_p$.
• The analysis leverages properties of $p$-adic exponential functions, which are only defined for $|x|_p \leq 1/p$, and their norm behavior $|\exp_p(x)|_p = 1$.

Experimental results

Research questions

• RQ1 Under what conditions on $p$ and $k$ does a $p$-adic splitting Gibbs measure exist for the hard core model on a Cayley tree?
• RQ2 What is the structure of $p$-adic Gibbs measures—specifically, are they only translational invariant, or do periodic measures (e.g., period two) also exist?
• RQ3 For $k=2$, what values of $p$ and $\lambda$ allow the existence of non-translation-invariant $p$-adic Gibbs measures, and how many such measures exist?
• RQ4 When is a $p$-adic Gibbs measure bounded, and how does this depend on $p$?
• RQ5 How do the structural properties of $p$-adic Gibbs measures differ fundamentally from those of real Gibbs measures?

Key findings

• A $p$-adic splitting Gibbs measure exists for the hard core model on a Cayley tree of order $k$ only if $p$ divides $2^k - 1$.
• If $p \mid 2^k - 1$ and $p \nmid k+2$, then there exists a unique translational invariant $p$-adic Gibbs measure.
• The model admits only translational invariant and period-two (chess-board) periodic $p$-adic Gibbs measures.
• For $k=2$, a $p$-adic splitting Gibbs measure exists if and only if $p=3$, and in this case, there are exactly two periodic (non-translation-invariant) Gibbs measures when $\lambda$ lies in a $p$-adic ball of radius $1/27$.
• A $p$-adic Gibbs measure is bounded if and only if $p \neq 3$, with the measure being unbounded when $p=3$.
• The existence of periodic $p$-adic Gibbs measures for $p \geq 7$ requires $p \mid 2^k - 1$ and $p \mid k - 2$, while for $p=2,3,5$, the condition is necessary but not necessarily sufficient.