  We consider rapidly mixing dynamical systems and link the decay of the
shortest distance between multiple orbits with the generalized fractal
dimension. We apply this result to multidimensional expanding maps and extend
it to the realm of random dynamical systems. For random sequences, we obtain a
relation between the longest common substring between multiple sequences and
the generalized R\'enyi entropy. Applications to Markov chains, Gibbs states
and the stochastic scrabble are given.
