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 $W_n^{\beta }$ tends to zero) from the weak disorder regime (in which $W_n^{\beta }$ converges to a nontrivial limit). The other, ${\bar{\beta }}_c$, delimits the very strong disorder regime (in which $W_n^{\beta }$ 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.