  We study the minimal distance between two orbit segments of length n, in a
random dynamical system with sufficiently good mixing properties. This problem
has already been solved in non-random dynamical system, and on average in
random dynamical systems (the so-called annealed version of the problem): it is
known that the asymptotic behavior for this question is given by a
dimension-like quantity associated to the invariant measure, called its
correlation dimension (or R{\'e}nyi entropy). We study the analogous quenched
question, and show that the asymptotic behavior is more involved: two
correlation dimensions show up, giving rise to a non-smooth behavior of the
associated asymptotic exponent.
