Strong disorder and very strong disorder are equivalent for directed polymers

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