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.
Categories: cond-mat.stat-mech cond-mat.mtrl-sci