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

We analyze, via Imry-Ma scaling arguments, the strong disorder phases that exist in low dimensions at all temperatures for directed polymers and interfaces in random media. For the uncorrelated Gaussian disorder, we obtain that 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 $R \sim L^{\nu}$ with $\nu=(3-d)/(4-d)>1/2$ for the wandering of the best favorable tube available. The corresponding free-energy then scales as $F \sim L^{\omega}$ with $\omega=2 \nu-1$ and the left tail of the probability distribution 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 $R_S \sim L^{1/3}$ is known as the typical distance between two replicas in the Bethe Ansatz wave function. We then extend our approach to correlated disorder in transverse directions with exponent $\alpha$ and/or to manifolds in dimension $D+d=d_{t}$ with $0<D<2$. The strategy of being both confined and superdiffusive is still optimal for decaying correlations ($\alpha<0$), whereas it is not for growing correlations ($\alpha>0$). In particular, for an interface of dimension $(d_t-1)$ in a space of total dimension $5/3<d_t<3$ with random-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 an algebraic tail $1/V^{1+\mu}$ in the disorder distribution, and obtain various regimes in the $(\mu,d)$ plane.