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 l₂ steps right, left, up and down. We aim, in particular, at the algebraic area generating function $Z_{m_1,m_2,l_1,l_2}\left(Q\right)$ evaluated at $Q=e^{\frac{2i\pi }{q}},$ 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 $Z_{m,0,l,0}\left(e^{\frac{2i\pi }{q}}\right)$ in terms of the cyclic sieving phenomenon is illustrated. Then, an expression for $Z_{m_1,m_2,l_1,l_2}(-1),$ 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. (paper)