[Paper Review] An extremality of the translation-invariant Gibbs measure for the HC-model on a Cayley tree

U. A. Rozikov, Р. М. Хакимов|arXiv (Cornell University)|Oct 15, 2016

Theoretical and Computational Physics1 references3 citations

TL;DR

This paper establishes the extremality of the translation-invariant Gibbs measure for the hard-core (HC) model on a Cayley tree of degree $k o rac{1}{k}$. It proves that for $k o 2$, the Gibbs measure $ u^*$ is extremal (i.e., not a nontrivial convex combination of other Gibbs measures) when the fugacity $ u$ is below a critical threshold $ u_* = u_*(k)$, derived from a nonlinear equation involving $k$ and a root $t_* o 1$ as $k o igcirc$. The result confirms the uniqueness of the Gibbs measure in a non-trivial regime, extending prior work on phase transitions and extremality in statistical mechanics on trees.

ABSTRACT

In this paper we study the extremality of translation-invariant Gibbs measure for the HC-model on a Cayley tree. It is known that for this model the translation-invariant measure is unique. We give a new proof of this statement and found regions of the extremality of this measure on the Cayley tree of order k.

Motivation & Objective

• To determine conditions under which the translation-invariant Gibbs measure for the hard-core model on a Cayley tree is extremal.
• To establish the existence of a critical fugacity threshold $ u_*(k)$ below which the Gibbs measure is extremal.
• To analyze the phase transition behavior of the hard-core model on Cayley trees using recursive equations and spectral methods.
• To extend previous results on extremality and uniqueness of Gibbs measures in statistical mechanics on tree graphs.

Proposed method

• Uses a recursive equation for the ratio $z_x' = z_{1,x}/z_{0,x}$, derived from the Gibbs measure condition, leading to $z = (1 + u z)^{-k}$.
• Analyzes the fixed-point equation $z = (1 + u z)^{-k}$ to determine the existence and number of solutions, corresponding to different Gibbs measures.
• Applies the Dobrushin uniqueness criterion via the quantities $ u( ext{measure})$ and $ u( ext{mixing})$, requiring $k u( ext{measure}) u( ext{mixing}) < 1$.
• Reduces the extremality condition to a polynomial inequality in $t = z^{1/k}$: $t^{k+1} - k t^2 + (2k - 1)t - k + 1 > 0$.
• Solves the inequality to define the critical threshold $ u_* = u_*(k)$ as the largest $ u$ for which the inequality holds for $t o (t_*, 1)$, where $t_*$ is the smallest root of the polynomial.
• Uses asymptotic analysis to show $ u_*(k) o 0$ as $k o igcirc$, and computes $ u_*(k)$ numerically for small $k$.

Experimental results

Research questions

• RQ1 For which values of the fugacity $ u$ is the translation-invariant Gibbs measure for the hard-core model on a Cayley tree of degree $k$ extremal?
• RQ2 What is the critical threshold $ u_*(k)$ below which the Gibbs measure is unique and extremal?
• RQ3 How does the extremality threshold $ u_*(k)$ behave as the tree degree $k$ increases?
• RQ4 Does the extremality condition derived from the Dobrushin uniqueness criterion hold for all $k \geq 2$?
• RQ5 What is the relationship between the fixed-point equation $z = (1 + u z)^{-k}$ and the extremality of the Gibbs measure?

Key findings

• The Gibbs measure $ u^*$ is extremal if and only if $ u < u_*(k)$, where $ u_*(k)$ is defined by the root $t_* o 1$ of the polynomial $t^{k+1} - k t^2 + (2k - 1)t - k + 1 = 0$.
• For $k = 2$, $ u_*(2) o 7.159$, and $ u_*(k)$ decreases with $k$, approaching 0 as $k \to igcirc$.
• The critical threshold $ u_{ ext{cr}}(k) = k^k / (k - 1)^{k+1}$ marks the onset of non-uniqueness; $ u_*(k) > u_{ ext{cr}}(k)$ for all $k \geq 2$.
• As $k \to igcirc$, the extremality threshold $ u_*(k) \to 0$, indicating that extremality is lost in the limit of high-degree trees.
• Numerical values show $ u_*(3) \approx 3.771$, $ u_*(4) \approx 2.745$, $ u_*(5) \approx 2.242$, and $ u_*(18) \approx 1.015$, with $ u_*(3000) \approx 0.250$.
• The extremality condition is equivalent to the inequality $t^{k+1} - k t^2 + (2k - 1)t - k + 1 > 0$ for $t \in (t_*, 1)$, where $t_*$ is the smallest root in $(0,1)$.