Solvable Models of Random Hetero-Polymers at Finite Density: I. Statics

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.