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)* similar to L-alpha. For large values its distribution obeys logP(h(L)*) similar to -A(h(L)*/L-alpha)(a). In the early-time regime where the roughness grows as t(beta), we find h(L)* similar to t(beta)[1nL - (beta/alpha) 1nt + 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-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.