Interface in a one-dimensional Ising spin system

In this paper we will be studying the interface in a one-dimensional Ising spin system with a ferromagnetic Kac potential γJ(γ|r|). Below the critical temperature, when γ tends to 0, two distinct thermodynamic phases with different magnetizations appear. We will see that the local magnetization converges to one of these two values. On intervals of length γ−k the local magnetization will stay almost constant, but on longer intervals interfaces take place between different phases. We prove first a large deviation principle and apply Friedlin and Wentzell theory to estimate the position where the first interface appears.