Log‐Sobolev inequality for near critical Ising models

Abstract

AbstractFor 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 when .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.