  We study the directed polymer model in a bounded environment in weak disorder
but without $L^2$-boundedness, specifically the speed of homogenization for the
field $(W_n^{0,x})_{x\in\mathbb Z^d}$, where $W_n^{0,x}$ denotes the associated
martingale for the polymer starting from $x$. We show that a suitably
re-centered spatial average over a set of diameter $n^{1/2}$ convergence to
zero at rate $n^{-\xi+o(1)}$, where the exponent is an explicit function of the
inverse temperature $\beta$.
