Anomalous roughness, localization, and globally constrained random walks

The scaling properties of a random walker subject to the global constraint that it needs to visit each site an even number of times are determined. Such walks are realized in the equilibrium state of one-dimensional surfaces that are subject to dissociative-dimer-type surface dynamics. Moreover, they can be mapped onto unconstrained random walks on a random surface, and the latter corresponds to a non-Hermitian random free fermion model that describes electron localization near a band edge. We show analytically that the dynamic exponent of this random walk is [formula presented] in spatial dimension d. This explains the anomalous roughness, with exponent [formula presented] in one-dimensional equilibrium surfaces with dissociative-dimer-type dynamics.