Local limit Theorem for Directed polymers in a random environment in the full weak disorder regime

For directed polymers in random environment, we provide a short and mostly self-contained proof of the local limit theorem in the full weak disorder phase, including the critical inverse temperature $β_c$. That is, the point-to-point partition function is well approximated by the product of two independent point-to-line partition functions (one starting from each endpoint of the system) multiplied by the simple random walk transition kernel. The approximation is valid in the $L^p$ sense for an optimal range of $p$. The result is new at $β=β_c$ and for $p>2$, and in all cases the proof is significantly simpler, technically and conceptually, than what can be found in the literature. As a consequence of the result we establish an invariance principle for the polymer measure with fixed end-point, valid for all $β\in [0,β_c]$.