Solvable lattice gas models of random hetero-polymers at finite density: I. Statics

We introduce infinity -dimensional lattice gas versions of three common mod els of random heteropolymers, in which both the polymer density and the den sity of the polymer-solvent mixture are finite. These solvable models give valuable insight into the problems related to the (quenched) average over t he randomness in statistical mechanical models of proteins, without having to deal with the hard geometrical constraints occurring in finite-dimension al models. Our exact solution, which is specific to the infinity -dimension al case, is compared to the results obtained by a saddle-point analysis and by the grand ensemble approach, both of which can also be applied to model s of finite dimension. We find, somewhat surprisingly, that the saddle-poin t analysis can lead to qualitatively incorrect results.