Exact Analysis of Level-Crossing Statistics for (d+1)-Dimensional Fluctuating Surfaces

Abstract

We carry out an exact analysis of the average frequency $ν_{αx_i}^+$ in the direction $x_i$ of positive-slope crossing of a given level $α$ such that, $h({\bf x},t)-\bar{h}=α$, of growing surfaces in spatial dimension $d$. Here, $h({\bf x},t)$ is the surface height at time $t$, and $\bar{h}$ is its mean value. We analyze the problem when the surface growth dynamics is governed by the Kardar-Parisi-Zhang (KPZ) equation without surface tension, in the time regime prior to appearance of cusp singularities (sharp valleys), as well as in the random deposition (RD) model. The total number $N^+$ of such level-crossings with positive slope in all the directions is then shown to scale with time as $t^{d/2}$ for both the KPZ equation and the RD model.