On the algebraic area of lattice walks and the Hofstadter model

Abstract

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.

17 pages, 2 figures

References in corpus (2)

• From lattice Quantum Electrodynamics to the distribution of the algebraic areas enclosed by random walks on $Z^2$
• Statistics of two-dimensional random walks, the "cyclic sieving phenomenon" and the Hofstadter model

Cited by in corpus (5)

• The algebraic area of closed lattice random walks
• SYK model with an extra diagonal perturbation: phase transition in the eigenvalue spectrum
• Hofstadter point spectrum trace and the Almost Mathieu operator
• Lattice random walks and quantum A-period conjecture
• On Thouless bandwidth formula in the Hofstadter model