[Paper Review] The weak limit of Ising models on locally tree-like graphs

Andrea Montanari, Elchanan Mossel+1 more|ArXiv.org|Dec 3, 2009

Theoretical and Computational Physics11 references4 citations

TL;DR

This paper establishes that the Ising model on locally tree-like graphs, such as regular graphs with diverging girth or expanders, converges locally in distribution to a symmetric mixture of plus and minus boundary condition Gibbs measures on the infinite k-regular tree. For zero external field (B=0), the weak limit is exactly the average of the + and − measures, and conditioning on positive magnetization yields convergence to the + measure, revealing a phase transition in the absence of external fields.

ABSTRACT

We consider the Ising model with inverse temperature beta and without external field on sequences of graphs G_n which converge locally to the k-regular tree. We show that for such graphs the Ising measure locally weak converges to the symmetric mixture of the Ising model with + boundary conditions and the - boundary conditions on the k-regular tree with inverse temperature β. In the case where the graphs G_n are expanders we derive a more detailed understanding by showing convergence of the Ising measure condition on positive magnetization (sum of spins) to the + measure on the tree.

Motivation & Objective

• To characterize the weak limit of Ising models on sequences of locally tree-like graphs, particularly those converging to the k-regular tree.
• To resolve the open problem of the weak limit when there is no external field (B=0), which is the most complex case due to symmetry and phase coexistence.
• To show that conditioning on positive magnetization leads to convergence to the + measure on the tree, providing a refined understanding of phase behavior in the absence of external fields.
• To establish local weak convergence results for Ising measures on expanders and locally tree-like graphs, using tools from statistical physics and probability theory.

Proposed method

• The authors use local weak convergence of graphs to the k-regular tree, defined via the neighborhood distribution of a uniformly random vertex.
• They define local weak convergence of Ising measures by examining the distribution of spin configurations in neighborhoods of a random vertex as the graph size grows.
• The key technique involves analyzing conditional expectations and variances of local functions under the Ising measure, particularly using the conditional expectation of spin functions given the spins outside a ball of radius r.
• They prove decay of covariance by showing that the variance of conditional expectations vanishes as r→∞, leveraging the extremality of the + and − Gibbs measures on the tree.
• The proof relies on coupling arguments and total variation bounds between finite-graph measures and the infinite-tree measures.
• They use the fact that for expanders, the probability that two random vertices are close decays with graph size, enabling control over long-range correlations.

Experimental results

Research questions

• RQ1 What is the weak limit of the Ising measure on locally tree-like graphs with zero external field (B=0)?
• RQ2 How does the Ising measure behave when conditioned on positive magnetization in the absence of external fields?
• RQ3 Does the local weak limit of Ising models on expanders converge to a single Gibbs measure or a mixture?
• RQ4 Can the symmetric mixture of + and − boundary condition measures on the infinite tree be obtained as the weak limit of Ising models on finite graphs with B=0?
• RQ5 What role does the girth or expansion property of the graph play in the convergence of the Ising measure?

Key findings

• For any β ≥ 0 and B = 0, the Ising measure on locally tree-like graphs converges weakly to the symmetric mixture (1/2)ν⁺ + (1/2)ν⁻ on the k-regular tree.
• When the graphs are expanders and the Ising measure is conditioned on positive magnetization (∑xᵢ > 0), it converges weakly to the + boundary condition measure ν⁺ on the tree.
• The same convergence holds for the negative magnetization case, converging to ν⁻.
• The convergence is local and holds for almost all vertices in the graph, meaning the distribution of spins in a fixed neighborhood of a random vertex converges to the corresponding tree measure.
• The variance of conditional expectations of local functions decays to zero as the radius r increases, which implies decay of long-range correlations and enables the convergence proof.
• The extremality of the ν⁺ and ν⁻ measures on the tree is crucial in showing that the conditional expectations converge to constants, which drives the variance decay.