# Log-Sobolev inequality for near critical Ising models

arXiv: https://arxiv.org/abs/2202.02301 · HTML: https://mlsanity.com/paper/2202.02301/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Feb 2022.

## Built on by (most cited first, 2 of 2)

- [Hierarchic Flows to Estimate and Sample High-dimensional Probabilities](https://mlsanity.com/paper/2405.03468/) (2024)
- [Functional Stochastic Localization](https://mlsanity.com/paper/2602.03999/) (2026)

## Abstract (arXiv)

For general ferromagnetic Ising models whose coupling matrix has bounded spectral radius, we show that the log-Sobolev constant satisfies a simple bound expressed only in terms of the susceptibility of the model. This bound implies very generally that the log-Sobolev constant is uniform in the system size up to the critical point (including on lattices), without using any mixing conditions. Moreover, if the susceptibility satisfies the mean-field bound as the critical point is approached, our bound implies that the log-Sobolev constant depends polynomially on the distance to the critical point and on the volume. In particular, this applies to the Ising model on subsets of mathbbZd when d>4. The proof uses a general criterion for the log-Sobolev inequality in terms of the Polchinski (renormalisation group) equation, a recently proved remarkable correlation inequality for Ising models with general external fields, the Perron--Frobenius theorem, and the log-Sobolev inequality for product Bernoulli measures.