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

TL;DR

This paper provides an exact analysis of level-crossing statistics for growing surfaces in multiple dimensions, focusing on the KPZ equation without surface tension and the random deposition model, revealing a universal scaling law.

Contribution

It derives an exact expression for the average frequency of positive-slope level crossings in multi-dimensional surface growth models, highlighting a universal $t^{d/2}$ scaling law.

Findings

Level-crossing frequency scales as $t^{d/2}$ in both models.

Exact analytical results for KPZ without surface tension.

Universal behavior before cusp singularity formation.

Abstract

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.