Localization transition for generalized directed polymers

Let the Hamiltonian be convex and let its associated Lagrangian have superlinear growth. A generalized directed polymer is the polymer process associated with the corresponding random Hamilton–Jacobi dynamics. Generalized polymer localization conjecture. In dimensions $d=1$ and $d=2$, the generalized directed polymer is localized for every such convex Hamiltonian. In dimensions $d\geq3$, it undergoes a transition from diffusive behaviour for small forcing potentials to localization for large forcing potentials. This generalizes the proposed quadratic-Hamiltonian localization picture; the source gives no resolution for general Hamiltonians.