Log-Sobolev inequality for near critical Ising models

TL;DR

The paper derives a bound on the log-Sobolev constant for ferromagnetic Ising models with bounded spectral radius, expressing 1/\\gamma_{\\beta,h} solely in terms of the zero-field susceptibility via $1/\\gamma_{\\beta,h} \le 1/2 + \int_0^{\\beta} e^{2\int_0^t \\chi_s ds}\, dt$. This yields uniform-in-volume log-Sobolev inequalities in the high-temperature regime without relying on mixing conditions, and, under a mean-field bound on the susceptibility, implies polynomial scaling of the bound near criticality and with system size, including Ising models on subsets of $\\mathbb{Z}^d$ with $d>4$. The approach combines a Polchinski renormalisation-group criterion, a powerful correlation inequality, the Perron–Frobenius theorem, and the log-Sobolev inequality for product Bernoulli measures, together with a decomposition into an infinite-temperature part and a renormalised continuous part. These results advance understanding of near-critical convergence to equilibrium for Ising systems and extend beyond geometrically mixing settings.

Abstract

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 $\mathbb{Z}^d$ 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.

Theorems & Definitions (4)

• theorem 1
• corollary 1
• remark 1
• proof : Proof of Theorem \ref{['thm:ising']}