One-point function of the four-vertex model

We consider the four-vertex model on a finite domain of the square lattice with the so-called scalar-product boundary conditions. It can be described in terms of non-intersecting lattice paths which are additionally restricted in their propagation in one of the two spacial directions. We compute the one-point function measuring the probability to obtain a path on a given lattice edge. We also relate this function with another one-point function which can be regarded as a local anti-ferroelectric order parameter.