This work did not investigate the logarithmic displacements of lattice sites (which are difficult to quantify for a finite system size), but the magnitude of the mean squared displacement of the particles as function of time. This way, the displacements are not analysed in space but in the time domain. The theoretical background is given by D. Cassi, as well as F. Merkl and H. Wagner. This work analyses the recurrence probability of random walks and spontaneous symmetry breaking in various dimensions. The finite recurrence probability of a random walk in one and two dimension shows a dualism to the lack of perfect long-range order in one and two dimensions, while the vanishing recurrence probability of a random walk in 3D is dual to existence of perfect long-range order and the possibility of symmetry breaking.