  We consider a lattice gas with quenched impurities or `quenched-annealed
binary mixture' on the Bethe lattice. The quenched part represents a porous
matrix in which the (annealed) lattice gas resides. This model features the 3
main factors of fluids in random porous media: wetting, randomness and
confinement. The recursive character of the Bethe lattice enables an exact
treatment, whose key ingredient is an integral equation yielding the
one-particle effective field distribution. Our analysis shows that this
distribution consists of two essentially different parts. The first one is a
continuous spectrum and corresponds to the macroscopic volume accessible to the
fluid, the second is discrete and comes from finite closed cavities in the
porous medium. Those closed cavities are in equilibrium with the bulk fluid
within the grand canonical ensemble we use, but are inaccessible in real
experimental situations. Fortunately, we are able to isolate their
contributions. Separation of the discrete spectrum facilitates also the
numerical solution of the main equation. The numerical calculations show that
the continuous spectrum becomes more and more rough as the temperature
decreases, and this limits the accuracy of the solution at low temperatures.
