Disorder-induced roughening of diverse manifolds

Abstract

We propose a unified treatment of the roughening of manifolds by impurities in quenched correlated random media. Our perspective incorporates such apparently distinct problems as domain walls in dirty Ising magnets, biased walks upon random lattices, and flux creep in high-${\mathit{T}}_{\mathit{c}}$ materials. By means of generalized Imry-Ma arguments and a functional renormalization group (RG), we find new results, including the random-bond interfacial roughening exponent, ${\mathrm{\ensuremath{\zeta}}}_{\mathrm{RB}}$=2\ensuremath{\epsilon}/9, as well as estimates for the many-dimensional directed-polymer wandering index. This last quantity is also investigated via real-space RG methods, where we find, for example, ${\mathrm{\ensuremath{\zeta}}}_{2+1}$=0.602, in reasonable agreement with the functional RG value 3/5. Finally, since the Burgers equation permits translation of our directed-polymer results to the Eden cluster and ballistic deposition problems in higher dimensions, we can compare to the most recent computer simulations of these stochastic growth models. In particular, we address issues regarding the exponent conjectures that have been made and suggest the possibility of a finite upper critical dimension.