  By using a recently proposed probabilistic approach, we determine the exact
ground state of a class of matrix Hamiltonian models characterized by the fact
that in the thermodynamic limit the multiplicities of the potential values
assumed by the system during its evolution are distributed according to a
multinomial probability density. The class includes i) the uniformly fully
connected models, namely a collection of states all connected with equal
hopping coefficients and in the presence of a potential operator with arbitrary
levels and degeneracies, and ii) the random potential systems, in which the
hopping operator is generic and arbitrary potential levels are assigned
randomly to the states with arbitrary probabilities. For this class of models
we find a universal thermodynamic limit characterized only by the levels of the
potential, rescaled by the ground-state energy of the system for zero
potential, and by the corresponding degeneracies (probabilities). If the
degeneracy (probability) of the lowest potential level tends to zero, the
ground state of the system undergoes a quantum phase transition between a
normal phase and a frozen phase with zero hopping energy. In the frozen phase
the ground state condensates into the subspace spanned by the states of the
system associated with the lowest potential level.
