Diverse Manifolds in Random Media

We suggest a unifying perspective of various manifolds subject to quenched correlated disorder. The new picture incorporates seemingly disparate problems as domain walls in impurity-stricken Ising magnets and directed walks upon random lattices. Using Flory-style arguments and a functional nonlinear renormalization group, we obtain new results, including the random-bond interfacial roughening exponent ${\zeta }_{\mathrm{RB}}=\frac{2}{9}\epsilon$, where $\epsilon =5-d$, which we believe to be exact, as well as a conjecture for the (2+1)-dimensional directed polymer index ${\zeta }_{2+1}=\frac{30}{49}$.