Fixed-magnetization Ising on random graphs up to reconstruction

2025-11-20

AI Summary

This paper studies the ferromagnetic Ising model with fixed magnetization on random $d$-regular graphs ($d geq 3$) below the tree reconstruction threshold. Employing local weak convergence, Bethe free energy analysis, large deviation principles, and the theory of belief propagation fixed points, we rigorously establish: (1) convergence of the free energy density to the annealed free energy—even across the spinodal interval between metastable magnetization states; (2) a complete characterization of the magnetization large deviation rate function, confirming the Zdeborová–Boettcher conjecture; (3) that the sum of proportions of bichromatic edges converges to unity in both the antiferromagnetic regime and the zero-magnetization ferromagnetic regime; and (4) rapid mixing of Glauber dynamics from uniform initial configurations in subexponential time. These results unify analytical frameworks bridging statistical physics and the asymptotic behavior of random graph models.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

We study the fixed-magnetization ferromagnetic Ising model on random $d$-regular graphs for $dge 3$ and inverse temperature below the tree reconstruction threshold. Our main result is that for each magnetization $η$, the free energy density of the fixed-magnetization Ising model converges to the annealed free energy density, itself the Bethe free energy of an Ising measure on the infinite $d$-regular tree. Moreover, the fixed-magnetization Ising model exhibits local weak convergence to this tree measure. A key challenge to proving these results is that for magnetizations between the model's spinodal points, the limiting tree measure corresponds to an unstable fixed point of the belief propagation equations. As an application, we prove that the positive-temperature Zdeborová--Boettcher conjecture on max-cut and min-bisection holds up to the reconstruction threshold: on the random $d$-regular graph, the expected fraction of bichromatic edges in the anti-ferromagnetic Ising model plus the expected fraction of bichromatic edges in the zero-magnetization ferromagnetic Ising model equals $1+o(1)$. A second application is completely determining the large deviation rate function for the magnetization in the Ising model on the random regular graph up to reconstruction. Finally, we use the precise understanding of this rate function to show that the Glauber dynamics for the full Ising model on the random graph mixes in sub-exponential time from uniformly random initialization, well into the non-uniqueness regime where the worst-case initialization mixing time is exponentially slow.

Problem

• Analyzing fixed-magnetization Ising models on random regular graphs
• Proving convergence to tree measures below reconstruction threshold
• Establishing mixing time properties for Glauber dynamics

Innovation

• Using belief propagation equations for analysis
• Applying local weak convergence to tree measures
• Studying Glauber dynamics mixing times