Maximal Height Scaling of Kinetically Growing Surfaces

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: $h^{*}_{L} \sim L^{\alpha}$. For large values its distribution obeys $\log{P(h^{*}_{L})} \sim -A({h^{*}_{L}}/L^{\alpha})^{a}$, charaterized by the exponential-tail exponent $a$. In the early-time regime where the roughness grows as $t^{\beta}$, we find $h^{*}_{L} \sim t^{\beta}[\ln{L}-({\beta\over \alpha})\ln{t} + C]^{1/b}$ 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.