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Abstract We report on the first self-consistent multidimensional particle-in-cell numerical simulations of nonhomogeneous pair discharges in polar caps of rotation-powered pulsars. By introducing strong inhomogeneities in the initial plasma distribution in our simulations, we analyze the degree of self-consistently emerging coherence of discharges across magnetic field lines. In 2D, we study discharge evolution for a wide range of physical parameters and boundary conditions corresponding to both the absent and free escape of charged particles from the surface of a neutron star. We also present the results of the first 3D simulations of discharges in a polar cap with a distribution of the global magnetospheric current appropriate for a pulsar with 60° inclination angle. For all parameters, we find the coherence scale of pair discharges across magnetic field lines to be of the order of the gap height. We also demonstrate that the popular “spark” model of pair discharges is incompatible with the universally adopted force-free magnetosphere model: intermittent discharges fill the entire zone of the polar cap that allows pair cascades, leaving no space for discharge-free regions. Our findings disprove the key assumption of the spark model about the existence of isolated distinct discharge columns.