We assume that the energy of contact between residues of a peptide chain is governed by an interaction matrix and derive a number of relationships between this interaction matrix and the energy spectrum over the compact states of the peptide chain. If the random energy model (REM) with a fixed number of contacts is assumed, the energy spectrum for the compact states of a peptide is known to be Gaussian. This leads to clear relations between the Hamiltonian, the energy spectrum, and the probability of a random peptide folding to a native state. While these developments are of great theoretical interest, it is evident that structural predictions for real proteins require a more detailed Hamiltonian which distinguishes the different types of residue-residue contacts. Here we consider a Hamiltonian which takes the form of an energy matrix and which explicitly defines the energy of the different types of residue-residue contacts. Statistical conditions are discussed for the contact sets of the compact states which again lead to a Gaussian energy distribution as a limiting form for large numbers of contacts. As for the REM, a simple relation exists between the energy matrix and the resulting energy spectrum. This in turn leads to predictions relating the energy matrix and the probability of a native state, and we show how such predictions may be extended to the case where the number of contacts is variable over the set of compact states. We further give the form of the energy matrix that will maximize the probability of a native state when the individual interaction energies obey certain plausible constraints. While these results may be regarded as exact for the limiting Gaussian distributions, we discuss the approximate nature of the results in realistic cases.