Exactness of the Annealed and the Replica Symmetric Approximations for Random Heteropolymers

We study a heteropolymer model with random contact interactions introduced some time ago as a simplified model for proteins. The model consists of self-avoiding walks on the simple cubic lattice, with contact interactions between nearest neighbor pairs. For each pair, the interaction energy is an independent Gaussian variable with mean value $B$ and variance $\D^2$. For this model the annealed approximation is expected to become exact for low disorder, at sufficiently high dimension and in the thermodynamic limit. We show that corrections to the annealed approximation in the 3-d high temperature phase are small, but do not vanish in the thermodynamic limit, and are in good agreement with our replica symmetric calculations. Such corrections derive from the fact that the overlap between two typical chains is nonzero. We explain why previous authors had come to the opposite conclusion, and discuss consequences for the thermodynamics of the model. Numerical results were obtained by simulating chains up to length 1400 by means of the recent PERM algorithm, in the coil and molten globular phases, well above the freezing temperature.