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

Abstract

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.

19 pages

References in corpus (3)

• Fragility of the Free-Energy Landscape of a Directed Polymer in Random Media
• Extremal statistics in the energetics of domain walls
• Elastic manifolds in disordered environments: energy statistics

Cited by in corpus (8)

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• Universal and non-universal tails of distribution functions in the directed polymer and KPZ problems
• Random walks and polymers in the presence of quenched disorder
• Universality classes of the Kardar-Parisi-Zhang equation
• Scaling and universality in coupled driven diffusive models
• Universality class of replica symmetry breaking, scaling behavior, and the low-temperature fixed-point order function of the Sherrington-Kirkpatrick model
• Joint free energy distribution in the random directed polymer problem
• Directed Polymers and Interfaces in Disordered Media