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 βc. 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 Lᵖ sense for an optimal range of p. The result is new at β=βc 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∈ [0,βc].