  We introduce $\infty$-dimensional versions of three common models of random
hetero-polymers, in which both the polymer density and the density of the
polymer-solvent mixture are finite. These solvable models give valuable insight
into the problems related to the (quenched) average over the randomness in
statistical mechanical models of proteins, without having to deal with the hard
geometrical constraints occurring in finite dimensional models. Our exact
solution, which is specific to the $\infty$-dimensional case, is compared to
the results obtained by a saddle-point analysis and by the grand ensemble
approach, both of which canalso be applied to models of finite dimension. We
find, somewhat surprisingly, that the saddle-point analysis can lead to
qualitatively incorrect results.
