# Log-Sobolev inequality for near critical Ising and continuum φ4 measures

I will present results on Glauber dynamics of Ising models and continuum

φ4 measures.

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 ℤd when d greater than 4.

For continuum φ4 measures in d=2,3, we show that log-Sobolev constant is bounded uniformly in the lattice regularisation and in the volume under the optimal condition that the susceptiblity is bounded.

Both proofs use a general criterion for the log-Sobolev inequality in terms of the Polchinski (renormalisation group) equation (proved with T. Bodineau) and rely on a recently proved remarkable correlation inequality for Ising models with general external fields (proved by Ding-Song-Sun).

This is joint work with Benoit Dagallier.

### Affiliation

University of Cambridge