Coincidence of critical points for directed polymers for general environments and random walks

For the directed polymer in a random environment (DPRE), two critical inverse-temperatures can be defined. The first one, $\beta_c$, separates the strong disorder regime (in which the normalized partition function $W^{\beta}_n$ tends to zero) from the weak disorder regime (in which $W^{\beta}_n$ converges to a nontrivial limit). The other, $\bar \beta_c$, delimits the very strong disorder regime (in which $W^{\beta}_n$ converges to zero exponentially fast). It was proved previously that $\beta_c=\bar \beta_c$ when the random environment is upper-bounded for the DPRE based on the simple random walk. We extend this result to general environment and arbitrary reference walk. We also prove that $\beta_c=0$ if and only the $L^2$-critical point is trivial.