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

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