Solution of the Kasteleyn model on a half-plane

A study is made of the Kasteleyn model, which is the simplest model of the phase transition of two-dimensional crystalline systems into a homogeneous incommensurate phase. An exact solution of this model on the half-plane is obtained. A dependence of the thermodynamic properties of the system of domain walls on the distance to the boundary is found. At small deviations $\tau$ from the critical temperature, the density of the domain walls behaves as $\tau^{3/2}$ near the boundary and $\tau^{1/2}$ far from it.