[Paper Review] Description of all translation-invariant $p$-adic Gibbs measures for the Potts model on a Cayley tree

U. A. Rozikov, Otabek Khakimov|arXiv (Cornell University)|Feb 25, 2014

advanced mathematical theoriesMathematics5 references21 citations

TL;DR

This paper fully characterizes and counts all translation-invariant $p$-adic Gibbs measures (TIpGMs) for the $q$-state Potts model on a Cayley tree of order two. Using recursive boundary law analysis and $p$-adic dynamics, it proves that TIpGMs are bounded if and only if $q \notin p\mathbb{N}$, and provides exact counts depending on $q$, $p$, and the interaction parameter $\theta$, with up to $2^q - 1$ distinct measures when $\theta \notin \{1-q, 1+q\}$ and $q \notin p\mathbb{N}$. The results generalize previous real-valued findings to the $p$-adic setting.

ABSTRACT

Recently it was proved that usual (real) Potts model on a Cayley tree has up to $2^q-1$ translation-invariant Gibbs measures. This paper is devoted to description of translation- invariant $p$-adic Gibbs measures (TIpGMs) of the $p$-adic Potts model. In particular, for the Cayley tree of order two we give exact number of such measures. Mereover we give criterion of boundedness of TIpGMs.

Motivation & Objective

• To fully describe the set of translation-invariant $p$-adic Gibbs measures (TIpGMs) for the $q$-state Potts model on a Cayley tree of order two.
• To determine the exact number of TIpGMs under various conditions on $q$, $p$, and the interaction parameter $\theta$.
• To establish a criterion for boundedness of TIpGMs in terms of the $p$-adic norm of $q$ and $\theta$.
• To extend the understanding of phase transitions in $p$-adic statistical mechanics beyond the real-valued case.

Proposed method

• Analyzes the tree recursion of boundary laws (boundary fields) whose fixed points correspond to TIpGMs.
• Characterizes fixed points by the number of non-zero components in the boundary law vector.
• Applies $p$-adic analysis, including $p$-adic exponential and logarithm functions, to study convergence and boundedness.
• Uses the recurrence relation $Z_{n+1,h} = A_{n,h} Z_{n,h}$ to analyze the normalizing constant $Z_n$ and its $p$-adic norm.
• Applies $p$-adic Hensel's lemma and solvability conditions for quadratic equations in $\mathbb{Q}_p$ to determine existence of solutions.
• Accounts for symmetries in the model to avoid overcounting distinct Gibbs measures.

Experimental results

Research questions

• RQ1 How many translation-invariant $p$-adic Gibbs measures exist for the $q$-state Potts model on a Cayley tree of order two, and how does this number depend on $q$, $p$, and $\theta$?
• RQ2 Under what conditions on $q$ and $\theta$ are the TIpGMs bounded in the $p$-adic sense?
• RQ3 What is the precise structure of fixed points of the boundary law recursion, and how do they relate to distinct Gibbs measures?
• RQ4 How does the number of TIpGMs change when $q \in p\mathbb{N}$ versus $q \notin p\mathbb{N}$?
• RQ5 What role does the $p$-adic norm of $\theta - 1$ play in determining the existence and boundedness of TIpGMs?

Key findings

• TIpGMs are bounded if and only if $q \notin p\mathbb{N}$, as shown by the divergence of the normalizing constant $Z_n$ in the $p$-adic norm when $q \in p\mathbb{N}$.
• When $q \notin p\mathbb{N}$, there is exactly one TIpGM, so $\mathcal{N}_{TI} = 1$.
• For $q = p > 2$ and $\theta \notin \{1-q, 1+q\}$, the number of TIpGMs is $\mathcal{N}_{TI} = 2^q - 1$, matching the real case.
• When $\theta \in \{1-q, 1+q\}$, the number of TIpGMs reduces to $\mathcal{N}_{TI} = 2^{q-1}$.
• For $q = pn$ with $n \in \{2, p-1\}$ and $p > 2$, the number of TIpGMs is reduced by $2\sum_{s=1}^{[n/2]} \binom{pn}{ps}$ when $\theta \notin \{1-q, 1+q\}$, and by $\sum_{s=1}^{[n/2]} \binom{pn}{ps}$ when $\theta \in \{1-q, 1+q\}$.
• For $p = 2$ and $|q|_2 > 1/4$, only one TIpGM exists, so $\mathcal{N}_{TI} = 1$, and for $q = 4$, up to 15 TIpGMs exist depending on the existence of a $2$-adic square root in $\mathbb{Q}_2$.