Exponential Relaxation of Glauber Dynamics with Some Special Boundary Conditions

Abstract.

We consider attractive finite-range Glauber dynamics and show that if a certain mixing condition is satisfied, then the system evolving on arbitrary subsets of the lattice, with appropriate boundary conditions, converges to equilibrium exponentially fast, in the uniform sense, uniformly over the subsets of the lattice. This result applies, for instance, to the ferromagnetic nearest neighbor Ising model in the so-called “Basuev region,” where complete analyticity is expected to fail.

Technically the result in this paper is an extension of a result of Martinelli and Olivieri, who proved that under a weaker form of mixing the infinite system approaches equilibrium exponentially fast.

Conceptually this paper may be seen as a step towards developing and exploiting a restricted notion of complete analyticity in which the boundary conditions, rather than the shapes of the regions under consideration, are being restricted.