  We analyse the propagation of particles in the narrow funnel of a thick
accretion disk. It is assumed that: (1) the funnel walls emit black body
radiation with temperature decreasing outwards; (2) the magnetic and electric
fields are longitudinal in the funnel. Such a scenario is consistent with
models in which a large potential drop is induced by a rotating massive black
hole threaded by a magnetic field (Blandford & Znajek 1977, Macdonald & Thorne
1982). The interaction of relativistic protons with thermal photons from the
funnel results in direct production of pairs and/or pions. We discuss the
relative importance of these processes for different conditions in the funnel
(temperature and electric field profiles). Injected pairs interact with thermal
photons in the Thomson or Klein-Nishina regime. Under some conditions, a pair
avalanche results, which we assume saturates in the production of stable
bunches, containing almost equal numbers of electrons and positrons. As a
consequence, highly collimated gamma-ray photons are produced. We obtain
gamma-ray spectra from our model in order to test its applicability to the
gamma-ray emitting AGNs recently reported by EGRET.
