Strong coupling probe for the Kardar-Parisi-Zhang equation

Abstract

We present an exact solution of the {\it deterministic} Kardar-Parisi-Zhang (KPZ) equation under the influence of a local driving force $f$. For substrate dimension $d \le 2$ we recover the well-known result that for arbitrarily small $f>0$, the interface develops a non-zero velocity $v(f)$. Novel behaviour is found in the strong-coupling regime for $d > 2$, in which $f$ must exceed a critical force $f_c$ in order to drive the interface with constant velocity. We find $v(f) \sim (f-f_c)^{\alpha (d)}$ for $f \searrow f_{c}$. In particular, the exponent $\alpha (d) = 2/(d-2)$ for $2<d<4$, but saturates at $\alpha(d)=1$ for $d>4$, indicating that for this simple problem, there exists a finite upper critical dimension $d_u=4$. For $d>2$ the surface distortion caused by the applied force scales logarithmically with distance within a critical radius $R_{c} \sim (f-f_{c})^{-\nu(d)}$, where $\nu(d) = \alpha (d)/2$. Connections between these results, and the critical properties of the weak/strong-coupling transition in the noisy KPZ equation are pursued.