In this paper, we consider directed polymers in random environment with
discrete space and time. For transverse dimension at least equal to 3, we prove
that diffusivity holds for the path in the full weak disorder region, i.e.,
where the partition function differs from its annealed value only by a
non-vanishing factor. Deep inside this region, we also show that the quenched
averaged energy has fluctuations of order 1. In complete generality (arbitrary
dimension and temperature), we prove monotonicity of the phase diagram in the
temperature
