  In many--body and other systems, the physics situation often allows one to
interpret certain, distinct states by means of a simple picture. In this
interpretation, the distinct states are not eigenstates of the full
Hamiltonian. Hence, there is an interaction which makes the distinct states act
as doorways into background states which are modeled statistically. The crucial
quantities are the overlaps between the eigenstates of the full Hamiltonian and
the doorway states, that is, the coupling coefficients occuring in the
expansion of true eigenstates in the simple model basis. Recently, the
distribution of the maximum coupling coefficients was introduced as a new,
highly sensitive statistical observable. In the particularly important regime
of weak interactions, this distribution is very well approximated by the
fidelity distribution, defined as the distribution of the overlap between the
doorway states with interaction and without interaction. Using a random matrix
model, we calculate the latter distribution exactly for regular and chaotic
background states in the cases of preserved and fully broken time--reversal
invariance. We also perform numerical simulations and find excellent agreement
with our analytical results.
