Constrained variational problem with applications to the Ising model

Roberto H. Schonmann, Senya B. Shlosman

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33 Citations (Scopus)


We continue our study of the behavior of the two-dimensional nearest neighbor ferromagnetic Ising model under an external magnetic field h, initiated in our earlier work. We strengthen further a result previously proven by Martirosyan at low enough temperature, which roughly states that for finite systems with ( - )-boundary conditions under a positive 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 value of the temperature. Here for every subcritical value of the temperature we show the existence of a critical value B0(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 the solution of the variational problem of finding the curve minimizing the Wulff functional, which curve is constrained to the unit square. Other tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field, extended to all temperatures below the critical one.

Original languageEnglish
Pages (from-to)867-905
Number of pages39
JournalJournal of Statistical Physics
Issue number5-6
Publication statusPublished - Jun 1996
Externally publishedYes


  • Boundary effects
  • Ising model
  • Large deviations
  • Wulff shape


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