On criticality for competing influences of boundary and external field in the Ising model

Consider the Gibb's measures μΛ(1/h),-,s (defined below) of the Ising model, in a box Λ(l/h) in Zd with side length 1/h, with external field s and negative boundary condition at a temperature T B2(T), the limit is μ+. This says that the negative boundary conditon dominates in the limit when B B2(T). The question, then, is whether there exists a critical value B0 = B0(T) = B1(T) = B2(T) for all T B0. In the case of d = 2, this question was completely solved by Schonmann and Shlosman (1996), using large deviation results and techniques. For higher dimensions, Greenwood and Sun (1997) ([GS] hereafter) proved the criticality of a certain value B0 for all T B0. In [Sch], the main results are about the relaxation time of a stochastic Ising model in relation to an external field h. He shows that the relaxation time blows up when h ↘ 0 as exp(λ/hd-1). In fact he obtains upper and lower bounds for λ = λ(T), which are derived from his B1(T), B2 (T) and his estimate of the spectral gap of the generator of the evolution. One might hope to obtain a critical value of λ using Schonmann's methods and the critical value B0. This indeed again gives bounds for λ but not a critical value. A reason is that estimation of the spectral gap is involved