On the algebraic area of lattice walks and the Hofstadter model

Stephane Ouvry 1 Stephan Wagner 2 Shuang Wu 1

Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2016, 49, pp.495205

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.

  • 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
  • 2. Department of Mathematics [Stellenbosch]