Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics

Abstract

We prove that the spectral gap of the Kawasaki dynamics shrink at the rate of 1/ L 2 for cubes of size L provided that some mixing conditions are satisfied. We also prove that the logarithmic Sobolev inequality for the Glauber dynamics in standard cubes holds uniformly in the size of the cube if the Dobrushin-Shlosman mixing condition holds for standard cubes.