On the algebraic area of lattice walks and the Hofstadter model
  We consider the generating function of the algebraic area of lattice walks, evaluated at a root of unity, and its relation to the Hofstadter model. In particular, we obtain an expression for the generating function of the n-th moments of the Hofstadter Hamiltonian in terms of a complete elliptic integral, evaluated at a rational function. This in turn gives us both exact and asymptotic formulas for these moments. 
Stephane Ouvry, Stephan Wagner, and Shuang Wu
math-ph cond-mat.stat-mech math.MP