On the algebraic area of lattice walks and the Hofstadter model

TL;DR

This paper explores the algebraic area of lattice walks at roots of unity and links it to the Hofstadter model, deriving formulas for moments of the Hofstadter Hamiltonian using elliptic integrals.

Contribution

It provides a novel connection between lattice walk generating functions and the Hofstadter model, including explicit formulas involving elliptic integrals.

Findings

Derived an expression for the generating function of the n-th moments of the Hofstadter Hamiltonian.

Obtained exact and asymptotic formulas for these moments.

Linked algebraic area generating functions to elliptic integrals evaluated at rational functions.

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.