  We are interested in the effect of Dirichlet boundary conditions on the nodal
length of Laplace eigenfunctions. We study random Gaussian Laplace
eigenfunctions on the two dimensional square and find a two terms asymptotic
expansion for the expectation of the nodal length in any square of side larger
than the Planck scale, along a denisty one sequence of energy levels. The proof
relies on a new study of lattice points in small arcs, and shows that the said
expectation is independent of the position of the square, giving the same
asymptotic expansion both near and far from the boundaries.
