Ising Model on Locally Tree-Like Graphs: Uniqueness of Solutions to Cavity Equations

In the study of Ising models on large locally tree-like graphs, in both rigorous and non-rigorous methods one is often led to understanding the so-called belief propagation distributional recursions and its fixed points. We prove that there is at most one non-trivial fixed point for Ising models with zero or certain random external fields. Previously this was only known for sufficiently “low-temperature” models. Our main innovation is in applying information-theoretic ideas of channel comparison leading to a new metric (degradation index) between binary-input-symmetric (BMS) channels under which the Belief Propagation (BP) operator is a strict contraction (albeit non-multiplicative). A key ingredient of our proof is a strengthening of the classical stringy tree lemma of Evans-Kenyon-Peres-Schulman (2000). Our result simultaneously closes the following 6 conjectures in the literature: 1) independence of robust reconstruction accuracy to leaf noise in broadcasting on trees; 2) uselessness of global information for a labeled 2-community stochastic block model, or 2-SBM; 3) optimality of local algorithms for 2-SBM under noisy side information; 4) uniqueness of BP fixed point in broadcasting on trees in the Gaussian (large degree) limit; 5) boundary irrelevance in broadcasting on trees; 6) characterization of entropy (and mutual information) of community labels given the graph in 2-SBM.