Rapid mixing within a phase for Ising models

Vasyl Paliy, Youngtak Sohn

Oct 16, 2026arXiv:2610.23777v1

math.PRcs.DSmath-ph

Scorecard· 16/16

7.0/10 impact

Technically strong, unifying framework with boundary-form stochastic localization and several concrete improvements/answers to open questions, but limited to large-signal/high-field regimes and overlapping concurrent work.

Abstract

We study Glauber dynamics for Ising measures conditioned to have a sufficiently large majority of $+ 1$ spins. Under a sparse spectral width condition on the interaction matrix, we derive a Poincaré inequality for the conditioned measures from a magnetization drift condition, and a modified logarithmic Sobolev inequality from a stronger, entropy weighted condition. As applications, we obtain approximate posterior sampling in total variation distance for $\mathbb{Z}_2$ synchronization at high enough signal-to-noise ratio using $O(n\log n)$ Glauber updates from a spectral initialization. For ferromagnetic Ising models on random $d$ -regular graphs, we prove rapid phase ordering down to inverse temperatures of order $1 / d$ , extending the temperature range of Gheissari, Sly, and Sohn (2025). Finally, we extend these inequalities to unconditioned Ising measures under an inward drift condition outside the conditioned region. This yields $O(n\log n)$ mixing of Glauber dynamics for the Sherrington-Kirkpatrick model at every fixed inverse temperature when the external field is sufficiently large.

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Scientific Impact Assessment: "Rapid mixing within a phase for Ising models" (Paliy & Sohn)

Core Contribution

The paper studies Glauber dynamics for Ising measures that have multiple phases and therefore mix exponentially slowly globally. It gives a general, non-monotone (not ferromagnetic-specific) framework for proving rapid mixing within a phase. Its main components are:

1. Sparse spectral width condition. Only principal submatrices of size at most n−M need small spectral width. This yields spectral independence (covariance bound 2/ε) for the core-conditioned measure. The proof uses the Percus/Lebowitz duplicate-replica trick, the Bauerschmidt–Bodineau/Chen–Eldan machinery, and negative dependence for sum-constrained product measures.

2. Stochastic localization with boundary forms. Conditioned product measures do not tensorize uniformly in the external field. The authors handle this by localizing on the ambient hypercube and adding quadratic and entropic boundary terms, which they show are supermartingales. This is the main technical innovation and is likely reusable.

3. Drift-based boundary absorption. Magnetization-drift conditions (H1)/(H1-E) let the boundary terms be absorbed into the Dirichlet form. This gives a Poincaré inequality and an MLSI of order 1/n on the core.

4. Lyapunov lifting. Foster–Lyapunov arguments extend the inequalities from the core to larger regions, and to the whole cube under inward drift.

Applications:

Z₂ synchronization posterior sampling in total variation with O(n log n) Glauber updates from spectral initialization, for all λ ≥ λ₀. This answers an open question raised in Montanari–Wu, Fan–Mei–Montanari and Bandeira–El Alaoui–Rödder.

Ferromagnetic Ising on random d-regular graphs, with phase ordering down to β ≍ 1/d. This removes the log d loss of Gheissari–Sly–Sohn and improves the required initial bias.

MLSI of order 1/n for the SK model at every β when the external field is large enough, which gives O(n log n) mixing.

Methodological Rigor

The proofs are detailed and appear internally consistent. The structure is clean: Proposition 2.1 (spectral independence), then 2.3 (localization with boundary), then 2.5 (absorption), then Theorem 1.2, then lifting in Theorem 1.3, then applications. Several steps are elegant:

the two-point log-Sobolev estimate used to compare the entropic boundary term with the quadratic one;

the Holley–Stroock perturbation, with oscillation 8Rω_M(J);

the exponential Lyapunov function for MLSI lifting.

The application proofs check the hypotheses with explicit random-matrix estimates (GOE norm bounds on submatrices, Friedman's theorem). The drift condition is shown to be necessary in a basic sense via the J=0 Cheeger example. Weak points:

constants are non-explicit, and the dependence exp(C‖J‖/ε) is crude (for synchronization this is exp(O(λ²)) in the MLSI constant, harmless only because λ is fixed);

the results need "sufficiently large" λ₀ and a large field, so they are far from sharp thresholds (for example, β < 1 for SK, or λ > 1 for synchronization);

no lower bounds or optimality of the initial bias are shown.

The AI-use statement is unusual: proofs were developed with substantial LLM assistance, though the authors say they verified everything. This does not affect the mathematics, but it may invite extra scrutiny of correctness until the results are independently checked.

Potential Impact

Theoretically, the paper shows how to get functional inequalities on a conditioned (non-product, non-monotone) state space with boundary, using localization schemes. It also gives a template, "core inequality + drift lifting", for within-phase mixing. This could be adopted for other multi-modal models: spin glasses with fields, Potts or block models, and Bayesian posteriors with planted structure. The synchronization result is a rare rigorous MCMC guarantee for a planted low-rank posterior in the high-SNR regime. It is complementary to the diffusion and AMP samplers (Montanari–Wu) and to concurrent work (Bandeira–El Alaoui–Rödder), which is closely related and overlaps in conditions. Its practical reach is limited: high-SNR synchronization, where spectral methods already largely solve estimation, and the high-field SK model, which is a weaker regime than the conjectured β < 1. For the regular-graph ferromagnet, it is a modest sharpening (removing log d) of a recent result.

Timeliness & Relevance

It is highly timely. Localization schemes, spectral independence and entropic independence are active areas. Phase ordering is the subject of a cluster of recent work (Gheissari–Sinclair, GSS25, Gheissari–Sly 2026). Mixing of Glauber dynamics for SK beyond β < 1/4 is a prominent open problem, and the high-field result is a partial step. The sparse spectral width condition is a natural relaxation of the full spectral width condition, which fails in low-temperature regimes.

Strengths

A unified framework that works for interactions of either sign, with no monotonicity.

A genuine technical innovation (boundary forms in stochastic localization) with a transparent explanation of why naive conditioning fails.

Several concrete applications, each improving or answering something in the literature.

Very clear exposition, with a roadmap, motivation and explicit hypotheses.

Limitations

The sparse spectral width condition is essentially equivalent, up to constants, to the concurrent Bandeira–El Alaoui–Rödder condition, which reduces the novelty margin.

The drift hypotheses are somewhat tailored and need field or SNR assumptions that exclude the interesting critical regimes.

Results are asymptotic, with "w.h.p." and unspecified constants. The SK corollary needs a large field and gives no insight at h=0.

Total-variation guarantees for synchronization rely on spectral initialization and the sign symmetry trick; there is no robustness analysis.

Because the framework is built from known components (Chen–Eldan localization, Lyapunov lifting, Holley–Stroock), the novelty is mostly in clever integration rather than a new paradigm.

Overall

This is a strong, technically solid probability and theoretical-CS paper. It is likely to be cited in the localization and phase-ordering literature, and its boundary-form localization may become a standard tool. The impact is mainly within mixing-time theory and high-dimensional Bayesian sampling theory, not across fields.

Rating:7/ 10

Significance 7Rigor 8Novelty 7Clarity 8

Generated Oct 19, 2026

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Scorecard· 16/16

7.0/10 impact