CONSTRAINED VARIATIONAL PROBLEM WITH APPLICATIONS TO THE ISING-MODEL

We continue our study of the behavior of the two-dimensional nearest n eighbor ferromagnetic Ising model under an external magnetic field ii, initiated in our earlier work. We strengthen further a result previou sly proven by Martirosyan at low enough temperature, which roughly sta tes that for finite systems with (-)-boundary conditions under a posit ive external field, the boundary effect dominates in the system if the linear size of the system is of order B/h with B small enough, while if B is large enough, then the external field dominates in the system. In our earlier work this result was extended to every subcritical val ue of the temperature. Here for every subcritical value oi the tempera ture we show the existence of a critical value B-0(T) which separates the two regimes specified above. We also find the asymptotic shape of the region occupied by the (+)-phase in the second regime, which turns out to be a ''squeezed Wulff shape.'' The main step in our study is t he solution of the variational problem of finding the curve minimizing the Wulff functional which curve is constrained to the unit square. O ther tools used are the results and techniques developed to study larg e deviations for the block magnetization in the absence of the magneti c field, extended to all temperatures below the critical one.