On criticality for competing influences of boundary and external field in the Ising model
- Greenwood, Priscilla E.
- Sun, Jiaming
2010 Mathematics Subject Classification
- 60K35 82B27
- Competing influences, Ising model, Gibbs measures
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 < Tc. It is well known that when s = 0, namely no external field, μΛ(l/h),-,0 converges weakly to the pure (－)-phase μ_ as h ↘ 0. And when s ≠ 0 is fixed, μΛ(l/h),-,s converges weakly to a measure μs which does not depend on the boundary conditons (Ellis (1985)). But if one lets the external field s decrease as h goes to zero in such a way that it competes with the negative boundary (in particular, if s = Bh), then one may obtain different limits in different ranges of B. This phenomenon of competing influences has been investigated by several authors. Martirosyan (1987) first proved that at low temperature T and with large B, the Gibb's measure μΛ(l/h),-,Bh converges weakly to the pure (＋)-phase μ+. Schonmann (1994) (referred to in the sequel as [Sch]) showed that at low temperature T, there are values B1(T) ≤ B2(T) such that when B < B1(T), μΛ(l/h),-,Bh converges weakly to μ_ and when B > B2(T), the limit is μ+. This says that the negative boundary conditon dominates in the limit when B < B1(T) whereas the small external field dominates when B > B2(T). The question, then, is whether there exists a critical value B0 = B0(T) = B1(T) = B2(T) for all T < Tc such that μΛ(l/h),-,Bh converges to μ_ when B < B0 and to μ+ when B > 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 < Tc, but only in terms of the convergence of average spins rather than in terms of weak convergence. This paper extends these results by showing that for low temperature and the same critical value B0, μΛ(l/h),-,Bh converges weakly to μ_ when B < B0 and to μ+ when B > 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.