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: ${h}_{L}^{*}\ensuremath{\sim}{L}^{\ensuremath{\alpha}}$. For large values its distribution obeys $\mathrm{log}P({h}_{L}^{*})\ensuremath{\sim}\ensuremath{-}A({h}_{L}^{*}{/L}^{\ensuremath{\alpha}}{)}^{a}$. In the early-time regime where the roughness grows as ${t}^{\ensuremath{\beta}}$, we find ${h}_{L}^{*}\ensuremath{\sim}{t}^{\ensuremath{\beta}}[\mathrm{ln}L\ensuremath{-}(\ensuremath{\beta}/\ensuremath{\alpha})\mathrm{ln}t+C{]}^{1/b}$, where either $b\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}a$ or $b$ is the corresponding exponent of the velocity distribution. These properties are derived from scaling and extreme-value arguments. They are corroborated by numerical simulations and supported by exact results for surfaces in 1D with the asymptotic behavior of a Brownian path.