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We analyze, via Imry-Ma scaling arguments, the strong disorder phases thatexist in low dimensions at all temperatures for directed polymers andinterfaces in random media. For the uncorrelated Gaussian disorder, we obtainthat the optimal strategy for the polymer in dimension $1+d$ with $0<d<2$involves at the same time (i) a confinement in a favorable tube of radius $R_S\sim L^{\nu_S}$ with $\nu_S=1/(4-d)<1/2$ (ii) a superdiffusive behavior $R \simL^{\nu}$ with $\nu=(3-d)/(4-d)>1/2$ for the wandering of the best favorabletube available. The corresponding free-energy then scales as $F \simL^{\omega}$ with $\omega=2 \nu-1$ and the left tail of the probabilitydistribution involves a stretched exponential of exponent $\eta= (4-d)/2$.These results generalize the well known exact exponents $\nu=2/3$, $\omega=1/3$and $\eta=3/2$ in $d=1$, where the subleading transverse length $R_S \simL^{1/3}$ is known as the typical distance between two replicas in the BetheAnsatz wave function. We then extend our approach to correlated disorder intransverse directions with exponent $\alpha$ and/or to manifolds in dimension$D+d=d_{t}$ with $0<D<2$. The strategy of being both confined andsuperdiffusive is still optimal for decaying correlations ($\alpha<0$), whereasit is not for growing correlations ($\alpha>0$). In particular, for aninterface of dimension $(d_t-1)$ in a space of total dimension $5/3<d_t<3$ withrandom-bond disorder, our approach yields the confinement exponent $\nu_S =(d_t-1)(3-d_t)/(5d_t-7)$. Finally, we study the exponents in the presence of analgebraic tail $1/V^{1+\mu}$ in the disorder distribution, and obtain variousregimes in the $(\mu,d)$ plane.Comment: 19 page

Directed polymers and interfaces in random media: Free-energy optimization via confinement in a wandering tube