  Random impedance networks are widely used as a model to describe plasmon
resonances in disordered metal-dielectric nanocomposites. In order to study
thin films, two-dimensional networks are often used despite the fact that such
networks correspond to a two-dimensional electrodynamics [J.P. Clerc et al, J.
Phys. A 29, 4781 (1996)]. In the present work, we propose a model of
two-dimensional systems with three-dimensional Coulomb interaction and show
that this model is equivalent to a planar network with long-range capacitive
connections between sites. In a case of a metal film, we get a known dispersion
$\omega \propto \sqrt{k}$ of plane-wave two-dimensional plasmons. In the
framework of the proposed model, we study the evolution of resonances with
decreasing of metal filling factor. In the subcritical region with metal
filling $p$ lower than the percolation threshold $p_c$, we observe a gap with
Lifshitz tails in the spectral density of states (DOS). In the supercritical
region $p>p_c$, the DOS demonstrates a crossover between plane-wave
two-dimensional plasmons and resonances associated with small clusters.
