Statistics of two-dimensional random walks, the "cyclic sieving phenomenon" and the Hofstadter model

TL;DR

This paper investigates the algebraic area distribution of 2D lattice random walks, explores connections to cyclic sieving phenomenon, and evaluates moments of the Hofstadter Hamiltonian, revealing new algebraic and geometric insights.

Contribution

It introduces a new algebraic area generating function at roots of unity, links it to cyclic sieving, and addresses moments of the Hofstadter Hamiltonian in specific cases.

Findings

Geometric interpretation of the generating function at roots of unity

Explicit expression for the generating function at Q=-1

Evaluation of Hofstadter Hamiltonian moments in the commensurate case

Abstract

We focus on the algebraic area probability distribution of planar random walks on a square lattice with $m_1$, $m_2$, $l_1$ and $l_2$ steps right, left, up and down. We aim, in particular, at the algebraic area generating function $Z_{m_1,m_2,l_1,l_2}(Q)$ evaluated at $Q=e^{2\i\pi\over q}$, 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 $Z_{m,0,l,0}(e^\frac{2i\pi}{q})$ 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's case, is proposed. Finally, the related problem of evaluating the n-th moments of the Hofstadter Hamiltonian in the commensurate case is addressed.