Statistics of two-dimensional random walks, the cyclic sieving phenomenon and the Hofstadter model

We focus on the algebraic area probability distribution of planar random walks on a square lattice with m(1), m(2), l(1) and l(2) steps right, left, up and down. We aim, in particular, at the algebraic area generating function Z(m1,m2,l1,l2) (Q) evaluated at Q = e2i pi/q, a root of unity, when both m(1) - m(2) and l(1) - l(2) are multiples of q. In the simple case of staircase walks, a geometrical interpretation of Z(m,0,l,0) (e2i pi/q) in terms of the cyclic sieving phenomenon is illustrated. Then, an expression for Z(m1,m2,l1,l2) (-1), which is relevant to the Stembridge case, is proposed. Finally, the related problem of evaluating the nth moments of the Hofstadter Hamiltonian in the commensurate case is addressed.