Ising model on preferential attachment models
Remco van der Hofstad, Rounak Ray
TL;DR
The paper analyzes the Ising model on affine preferential attachment graphs by leveraging local convergence to the Pólya point tree and belief-propagation methods. It derives the thermodynamic limit of the pressure, magnetisation, and internal energy in closed form, and identifies the inverse critical temperature β_c(m,δ) explicitly: β_c(m,δ) = atanh{ δ / [ 2( m(m+δ)+√(m(m−1)(m+δ)(m+δ+1)) ) ] } for δ>0, while β_c(m,δ)=0 for δ∈(-m,0]. The results connect phase transitions in the PA models to the percolation threshold on the PPT, providing a rigorous bridge between random graph structure and statistical mechanics on networks. The approach hinges on belief propagation on locally tree-like graphs and the PPT as the local limit, enabling exact thermodynamic characterisation in a broad random-graph setting with power-law-like degree distributions.
Abstract
We study the Ising model on affine preferential attachment models with general parameters. We identify the thermodynamic limit of several quantities, arising in the large graph limit, such as pressure per particle, magnetisation, and internal energy for these models. Furthermore, for $m\geq 2$, we determine the inverse critical temperature for preferential attachment models as $β_c(m,δ)=0$ when $δ\in(-m,0]$, while, for $δ>0$, $$β_c(m,δ)= {\rm atanh}\left\{ \fracδ{2\big( m(m+δ)+\sqrt{m(m-1)(m+δ)(m+δ+1)} \big)} \right\}~.$$ Our proof for the thermodynamic limit of pressure per particle critically relies on the belief propagation theory for factor models on locally tree-like graphs, as developed by Dembo, Montanari, and Sun. It has been proved that preferential attachment models admit the Pólya point tree as their local limit under general conditions. We use the explicit characterisation of the Pólya point tree and belief propagation for factor models to obtain the explicit expression for the thermodynamic limit of the pressure per particle. Next, we use the convexity properties of the internal energy and magnetisation to determine their thermodynamic limits. To study the phase transition, we prove that the inverse critical temperature for a sequence of graphs and its local limit are equal. Finally, we show that $β_c(m,δ)$ is the inverse critical temperature for the Pólya point tree with parameters $m$ and $δ$, using results from Lyons who shows that the critical inverse temperature is closely related to the percolation critical threshold. This part of the proof heavily relies on the critical percolation threshold for Pólya point trees established earlier with Hazra.