  Asymptotic properties of finitely generated subgroups of free groups, and of
finite group presentations, can be considered in several fashions, depending on
the way these objects are represented and on the distribution assumed on these
representations: here we assume that they are represented by tuples of reduced
words (generators of a subgroup) or of cyclically reduced words (relators).
Classical models consider fixed size tuples of words (e.g. the few-generator
model) or exponential size tuples (e.g. Gromov's density model), and they
usually consider that equal length words are equally likely. We generalize both
the few-generator and the density models with probabilistic schemes that also
allow variability in the size of tuples and non-uniform distributions on words
of a given length.Our first results rely on a relatively mild prefix-heaviness
hypothesis on the distributions, which states essentially that the probability
of a word decreases exponentially fast as its length grows. Under this
hypothesis, we generalize several classical results: exponentially generically
a randomly chosen tuple is a basis of the subgroup it generates, this subgroup
is malnormal and the tuple satisfies a small cancellation property, even for
exponential size tuples. In the special case of the uniform distribution on
words of a given length, we give a phase transition theorem for the central
tree property, a combinatorial property closely linked to the fact that a tuple
freely generates a subgroup. We then further refine our results when the
distribution is specified by a Markovian scheme, and in particular we give a
phase transition theorem which generalizes the classical results on the
densities up to which a tuple of cyclically reduced words chosen uniformly at
random exponentially generically satisfies a small cancellation property, and
beyond which it presents a trivial group.
