Heisenberg $XX0$ chain and random walks on a ring

We obtain and investigate mean values of the exponential of the centroid operator for the periodic Heisenberg $XX0$ chain on a ring. The generating function of directed lattice paths is obtained in terms of circulant matrices which leads to generalizations of the Ramus's identity. The two-time correlation function of the exponential of the centroid operator is expressed in terms of the Cauchy determinant and thus explicitly calculated.