Strong disorder and very strong disorder are equivalent for directed polymers

We show that if the normalized partition function $W_n^{\beta }$ of the directed polymer model on ${\mathbb{Z}}^d$ converges to zero, then it does so exponentially fast. This implies that there exists a critical value ${\beta }_c$ for the inverse temperature such that the normalized partition function has a non-degenerate limit for all $\beta \in [0,{\beta }_c]$ – weak disorder holds – while for $\beta \in ({\beta }_c,\mathrm{\infty })$ it converges exponentially fast to zero – very strong disorder holds. This solves a twenty-years-old conjecture formulated by Comets, Yoshida, Carmona and Hu. Our proof requires a technical assumption on the environment, namely, that it is bounded from above.