Large-Nexpansion of (4-ε)-dimensional oriented manifolds in random media

Abstract

The equilibrium statistical mechanics of a d-dimensional ‘‘oriented’’ manifold in an (N+d)-dimensional random medium are analyzed in d=(4-ε) dimensions. For N=1, this problem describes an interface pinned by impurities. For d=1, the model becomes identical to the directed polymer in a random medium. Here, we generalize the functional-renormalization-group method used previously to study the interface problem, and extract the behavior in the double limit ε small and N large, finding nonanalytic corrections in 1/N. For short-range disorder, the interface width scales as ω∼${\mathit{L}}^{\mathrm{\zeta }}$, with ζ=[ε/(N+4)]{1+(1/4e)$2^{\mathrm{-}\mathrm{u}\left[\right(\mathit{N}+2\left)2\mathrm{}\right]\left[\right(\mathit{N}+2)2/(\mathit{N}+4\left)\right]}$ [1-4/(N+2)+...]}. We also analyze the behavior for disorder with long-range correlations, as is appropriate for interfaces in random-field systems, and study the crossover between the two regimes.