Maximal Height Scaling of Kinetically Growing Surfaces

Abstract

The scaling properties of the maximal height of a growing self-affine surface with a lateral extent $L$ are considered. In the late-time regime its value measured relative to the evolving average height scales like the roughness: $h_L^{\ast }\sim L^{\alpha }$. For large values its distribution obeys $\log P(h_L^{\ast })\sim -A(h_L^{\ast }/L^{\alpha }{)}^a$, charaterized by the exponential-tail exponent $a$. In the early-time regime where the roughness grows as $t^{\beta }$, we find $h_L^{\ast }\sim t^{\beta }[\ln L-(\frac{\beta }{\alpha })\ln t+C{]}^{1/b}$ where either $b=a$ or $b$ is the corresponding exponent of the velocity distribution. These properties are derived from scaling and extreme-values arguments. They are corroborated by numerical simulations and supported by exact results for surfaces in 1D with the asymptotic behavior of a Brownian path.

One reference added. Minor stylistic changes in the abstarct and the paper. 4 pages, 3 figures

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