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

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

2005-08-07

arxiv.org/abs/cond-mat/0508180v2

JOURNAL OF STATISTICAL PHYSICS 124 (6): 1471-1490 (2006)

Physics

Condensed Matter

Statistical Mechanics

22 pages, 3 figures

Scientific paper

10.1007/s10955-006-9179-7

We carry out an exact analysis of the average frequency $\nu_{\alpha x_i}^+$ in the direction $x_i$ of positive-slope crossing of a given level $\alpha$ such that, $h({\bf x},t)-\bar{h}=\alpha$, 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.

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