Abstract

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, ${\overline{\beta }}_c$, delimits the very strong disorder regime (in which $W_n^{\beta }$ converges to zero exponentially fast). It was proved in earlier work that ${\beta }_c={\overline{\beta }}_c$ when the random environment is bounded above for the DPRE based on the simple random walk. We extend this result to general environments and an arbitrary reference walk. We also prove that ${\beta }_c=0$ if and only if the $L^2$ critical point is trivial.

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