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

Sergey Matveenko 1, 2 Stephane Ouvry 2

Journal of Physics A: Mathematical and Theoretical, Institute of Physics: Hybrid Open Access, 2014, 47, pp.185001

When random walks on a square lattice are biased horizontally to move solely to the right, the probability distribution of their algebraic area can be exactly obtained. We explicitly map this biased classical random system on a non hermitian Hofstadter-like quantum model where a charged particle on a square lattice coupled to a perpendicular magnetic field hopps only to the right. In the commensurate case when the magnetic flux per unit cell is rational, an exact solution of the quantum model is obtained. Periodicity on the lattice allows to relate traces of the Nth power of the Hamiltonian to probability distribution generating functions of biased walks of length N.

  • 1. Landau Institute for Theoretical Physics
  • 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques