Title: “Log-Sobolev inequality for near-critical Ising models”.

Abstract: For ferromagnetic Ising models, 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  ℤ^d when d greater than 4. The proof uses a relation of the log-Sobolev inequality to the
Polchinski (renormalisation group) equation (developed earlier with T. Bodineau), which I will explain, and also relies on a recently proved
remarkable correlation inequality for Ising models with general external
fields (proved by Ding-Song-Sun). Related methods apply to continuum models, but will not be discussed in detail in this talk. This is joint work with Benoit Dagallier.