The droplet in the tube: A case of phase transition in the canonical ensemble

Abstract

We consider the 2-dimensional Ising model with ferromagnetic nearest neighbour interaction at inverse temperatureβ. LetS _N =Σσ _t be the total magnetization inside anN×N square boxΛ,μ ^per_Λ be the Gibbs state inΛ with periodic b.c., andm(β) be the spontaneous magnetization. We show the existence of the limit

$$\psi (\varrho ) = \mathop {\lim }\limits_{N \to \infty } \left( { - \frac{1}{{\beta N}}} \right)\ln \mu _\Lambda ^{per} (S_N = [N\varrho ])$$

for |ϱ|<m(β), providedβ is large enough. It turns out that the quantityψ(ϱ) is closely related to the Wulf construction, and the dependence of the functionψ(ϱ) onϱ is singular.