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

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 $Q=e^{\frac{2ı\pi }{q}}$, 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.

13 pages, LaTeX 2e

References in corpus (1)

• Area distribution of two-dimensional random walks and non Hermitian Hofstadter quantum mechanics

Cited by in corpus (1)

• On the algebraic area of lattice walks and the Hofstadter model