We present a renewal theorem for random walks with "ultra heavy tails", which converge after rescaling to an explicit Levy process called the Dickman subordinator. Besides its own interest, such a renewal theorem is the key to a ne understanding of a challenging model in statistical mechanics, the directed polymer in random environment in the critical space dimension d=2, and of a singular stochastic PDE, the multiplicative Stochastic Heat Equation. We discuss the relations between these seemingly unrelated models, which
led to important recent progress. (Based on joint work with R. Sun and N. Zygouras)