Rapid mixing within a phase for Ising models
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 ℤ2 synchronization at high enough signal-to-noise ratio using O(nlog 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(nlog n) mixing of Glauber dynamics for the Sherrington-Kirkpatrick model at every fixed inverse temperature when the external field is sufficiently large.
Related
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BibTeX
@misc{paliy2026rapid,
title = {{Rapid mixing within a phase for Ising models}},
author = {Vasyl Paliy and Youngtak Sohn},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23777},
eprint = {2610.23777},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23777},
}
RIS
TY - UNPB
TI - Rapid mixing within a phase for Ising models
AU - Vasyl Paliy
AU - Youngtak Sohn
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23777
UR - https://arxiv.org/abs/2610.23777
AB - 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 ℤ_2 synchronization at high enough signal-to-noise ratio using O(nlog 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(nlog n) mixing of Glauber dynamics for the Sherrington-Kirkpatrick model at every fixed inverse temperature when the external field is sufficiently large.
ER -