The Phase Separation Line in the Two-Dimensional Ising Model*

We prove that, at low temperature, the line of separation between the two pure phases shows large fluctuations in shape. This implies the translation invariance of the correlation functions associated with some non translation invariant boundary conditions and should be a peculiarity of the dimensionality of the model. 1. The Line of Separation It has recently been conjectured that the surface of separation between two pure phases is, at low temperature and for short range potential models, rigid in the case of a 3-dimensional model and non rigid in 2-dimensional models [1, 2]. In this paper we prove the truth of the conjecture in the 2-dimensional Ising model. The precise meaning of what "surface of separation" and "rigid" mean will be given below and has already been discussed in the literature [3]. Let Ω be a N x N square lattice centered at the origin: let i — 1, 2,..., N 2 be a label for the center of each unit square composing Ω. We assume that on each site i e Ω is located a spin σ{ — ± 1 and that the energy of a spin configuration ρ = (σγ,..., σN2) is given by: H N W = i l > i f f j i X σi + i Σ σs (1.1) ϊedΩ \ed~ Ω where £ means, as usual, sum over the pairs of nearest neighbour < i j > couples of points in Ω and d Ω (d Ω) denote the points adjacent to the upper half (lower half) of the boundary dΩ of Ω. The physical meaning of (1.1) is that HN(σ) corresponds to the energy of a configuration of spins interacting through a nearest neighbour pair potential and, also, interacting with a set of external fixed spins adjacent * This work has been partially supported by the Consiglio Nazionale delle Ricerche (Gruppo Nazionale per ΓAnalisi Funzionale): CNR(GNAFA). 8 Commun math Phys., \ ol 27