Local limit theorem for directed polymers beyond the $L^{2}$-phase

We consider the directed polymer model in the weak disorder phase under the assumption that the partition function is L^{p} -bounded for some p>1+{2}/d . We prove that the point-to-point partition function can be approximated by two point-to-plane partition functions at the startpoint and endpoint, and in particular that it is L^{p} -bounded as well. Some consequences of this result are also discussed, the most important of which is a local limit theorem for the polymer measure. We furthermore show that the required L^{p} -boundedness holds for some range of \beta beyond the L^{2} -critical point, and in the whole interior of the weak disorder phase for environments with finite support.