Strong disorder and very strong disorder are equivalent for directed polymers

We show that if the normalized partition function W^β_n of the directed polymer model on ℤ^d converges to zero, then it does so exponentially fast. This implies that there exists a critical temperature β_c such that the renormalized partition function has a non-degenerate limit for all β∈ [0,β_c] – weak disorder holds – while for β∈ (β_c,∞) 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.