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: hL*~L^α. For large values its distribution obeys logP(hL*)~-A(hL*/L^α)ᵃ. In the early-time regime where the roughness grows as t^β, we find hL*~t^β[lnL-(β/α)lnt+C]1/b, where either b0ex0ex=0ex0exa 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.