Archive ouverte HAL – The algebraic area of closed lattice random walks

Stephane Ouvry 1 Shuang Wu 1

Stephane Ouvry, Shuang Wu. The algebraic area of closed lattice random walks. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2019, ⟨10.04098⟩. ⟨hal-02292208⟩

We propose a formula for the enumeration of closed lattice random walks of length $n$ enclosing a given algebraic area. The information is contained in the Kreft coefficients which encode, in the commensurate case, the Hofstadter secular equation for a quantum particle hopping on a lattice coupled to a perpendicular magnetic field. The algebraic area enumeration is possible because it is split in $2^{n/2-1}$ pieces, each tractable in terms of explicit combinatorial expressions.

  • 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques