Bound on the mass gap for finite volume stochastic ising models at low temperature

Abstract

We consider a sequence of finite volume Λ⊂Z d,d≧2, reversible stochastic Ising models in the low temperature regime and having invariant measures satisfying free boundary conditions. We show that associated with the models are random hitting times whose expectations, regarded as a function of Λ, grow exponentially in |Λ|(d-1)/d; moreover, the mass gaps for the models shrink exponentially fast in |Λ|(d-1)/d. A geometrical lemma is employed in the analysis which states that if a Peierls' contour is sufficiently small relative to the faces of Λ, then the fraction of the contour tangent to the faces is less than a constant smaller than one.