Domain growth, directed polymers, and self-organized criticality

The nature of the roughness of the growing surface modeled by the Kardar-Parisi-Zhang (KPZ) equation has been further studied by defining a ``domain'' structure for it. This was done by mapping the height h(x,t) of the surface onto a ``spin'' S(x,t)=sgn[h(x,t)-〈h(t)〉], where 〈h(t)〉 is the mean height of the growing surface at time (t). It was then found that the growth of the surface has useful analogies with the domain-coarsening process in nonequilibrium systems quenched into an ordered phase. Thus, in d=1+1, the average size 〈l〉 of spin domains grows as ${\mathit{t}}^{1/3}$ and the domain-size distribution P(l,t) is consistent with ${\mathit{l}}^{\mathrm{\ensuremath{-}}3/2}$${\mathit{f}}_{1}$(l/${\mathit{t}}^{2/3}$). We find that the autocorrelation function A(t)=〈S(x,t)S(x,0)〉 decays as ${\mathit{t}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$, with \ensuremath{\alpha}(d=1+1)=1.0\ifmmode\pm\else\textpm\fi{}0.08 and \ensuremath{\alpha}(d=2+1)=1.5\ifmmode\pm\else\textpm\fi{}0.1. The form of P(l,t) in higher dimensions and the connection between the spin autocorrelation function and the height autocorrelation function are discussed. For computational convenience, the KPZ equation was studied by transforming it to the problem of directed polymers in a random potential.