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

Abstract

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 obtained exactly (Mashkevich and Ouvry 2009 J. Stat. Phys. 137 71). We explicitly map this biased classical random system onto a non-Hermitian Hofstadter-like quantum model where a charged particle on a square lattice coupled to a perpendicular magnetic field hops only to the right. For the commensurate case, when the magnetic flux per unit cell is rational, an exact solution of the quantum model is obtained. The periodicity of the lattice allows one to relate traces of the Nth power of the Hamiltonian to probability distribution generating functions of biased walks of length N.