  Motivated by the search for best lattice sphere packings in Euclidean spaces
of large dimensions we study randomly generated perfect lattices in moderately
large dimensions (up to d=19 included). Perfect lattices are relevant in the
solution of the problem of lattice sphere packing, because the best lattice
packing is a perfect lattice and because they can be generated easily by an
algorithm. Their number however grows super-exponentially with the dimension so
to get an idea of their properties we propose to study a randomized version of
the algorithm and to define a random ensemble with an effective temperature in
a way reminiscent of a Monte-Carlo simulation. We therefore study the
distribution of packing fractions and kissing numbers of these ensembles and
show how as the temperature is decreased the best know packers are easily
recovered. We find that, even at infinite temperature, the typical perfect
lattices are considerably denser than known families (like A_d and D_d) and we
propose two hypotheses between which we cannot distinguish in this paper: one
in which they improve Minkowsky's bound phi\sim 2^{-(0.84+-0.06) d}, and a
competitor, in which their packing fraction decreases super-exponentially,
namely phi\sim d^{-a d} but with a very small coefficient a=0.06+-0.04. We also
find properties of the random walk which are suggestive of a glassy system
already for moderately small dimensions. We also analyze local structure of
network of perfect lattices conjecturing that this is a scale-free network in
all dimensions with constant scaling exponent 2.6+-0.1.
