Diffusion in a random catalytic environment, polymers in random media, and stochastically growing interfaces

We study a family of equivalent models which includes the polymer in a random medium, the stochastically growing interface with spatially random deposition of particles, and the diffusion in a random catalytic environment ${\partial }_t$ψ(x,t)=D${\nabla }^2$ψ+U(x)ψ. $U( x vec—) denotes a frozen Gaussian random potential of strength $u_0$ and correlation length a. The intrinsic length scales of the problem, $l_0$=(D/$u_0$ ${)}^{2/\left(4\mathrm{-}d\right)}$ and a, are both assumed to be small in comparison with the diffusion length √Dt and the system size L (L${<10\mathrm{}}^8$a). Flory-Imry-Ma–type arguments show that for dimensions d<4 the polymer of contour length t→∞ is both collapsed (ν=0) and localized, in agreement with previous results for a≪$l_0$ [Edwards and Muthukumar, J. Chem. Phys. 89, 2435 (1988); and Cates and Ball, J. Phys. (Paris) 49, 2009 (1988)]. The sample-to-sample variations of the polymer free-energy scale as $t^{\chi }$ with χ=1. For d>4, a collapsed, localized or a Gaussian, delocalized polymer is found for strong or weak disorder, respectively. The disorder becomes irrelevant for self-avoiding polymers. For growing interfaces, the roughness exponent χ/ν and dynamical exponent 1/ν are both equal to unity, but scaling is modified by logarithmic corrections.