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, that is, where the partition function differs from its annealed value only by a nonvanishing 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.
Francis Comets. Nobuo Yoshida. "Directed polymers in random environment are diffusive at weak disorder." Ann. Probab. 34 (5) 1746 - 1770, September 2006. https://doi.org/10.1214/009117905000000828